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 Grade 1 Measurement and Geometry
Measurement is explored in a very concrete manner. Before grade 1, students explore measurement using direct comparison. That is, they compare two objects directly to see which is longer, heavier, or holds more for length, mass, and capacity, respectively. The idea of a baseline for direct linear comparison (i.e., length) is established here. In grade 1, students build on these ideas with a focus on linear measurement using non-standard units rather than comparing two objects. Standard units can be uniform (e.g., standard paper clips) or non-uniform (e.g., hand width). The concepts of measuring using a baseline and iterating single units vs using multiple copies of a single unit are explored. In grade 2, students extend their understanding of non-standard units to standard units (e.g., centimetres).
Geometry is explored using visual (2D) and concrete (3D) materials. Before grade 1, students explore single attributes of 2D shapes and 3D objects through play and sorting. They explore drawing 2D shapes and building 3D objects. Using specific terminology to describe shapes and objects is not expected at this level. In grade 1, students begin to sort using a single attribute and explain their sorting rule. Students can also describe how multiple copies of one shape can compose to create a new shape. They can compare shapes and objects in their environment and use positional language to describe their location. In grade 2, students extend their sorting to include two attributes and explain their sorting rule. They continue to describe, create, and compare 2D shapes and 3D objects using math terminology, including knowing the names of shapes and objects. They can also identify 2D shapes within 3D objects.
Key Concepts
Direct measurement with non-standard units
Students explore non-standard units, including uniform and non-uniform units, ideas about baselines, repeating copies of a unit, and direct measurement using non-standard units.
Comparison of 2D Shapes and 3D objects
Students compare and sort 2D Shapes and 3D objects. They also discuss how shapes can be composed of other shapes.
Key Measurement and Geometry Concept 1: Direct measurement with non-standard units
Overview
Measurement is explored in a very concrete manner. Before grade 1, students explore measurement using direct comparison. That is, they compare two objects directly to see which is longer, heavier, or holds more for length, mass, and capacity, respectively. The idea of a baseline for direct linear comparison (i.e., length) is established here. In grade 1, students build on these ideas with a focus on linear measurement using non-standard units rather than comparing two objects. Standard units can be uniform (e.g., standard paper clips) or non-uniform (e.g., hand width). The concepts of measuring using a baseline and iterating single units vs using multiple copies of a single unit are explored. In grade 2, students extend their understanding of non-standard units to standard units (e.g., centimetres).
In developing ideas about direct measurement, it is useful to begin by reviewing direct linear comparison as a way of determining whether two objects are similar in length or which is longer/shorter. Doing so helps to establish a clear understanding of what constitutes a baseline. Lining up two objects end-to-end allows us to more easily compare two objects. This idea extends to using units. We want to stress the importance of beginning to measure from the baseline for consistent results.
Sometimes it is not practical to use direct linear comparison. For example, we wouldn’t want to unhinge a door to compare it to the length of a table. Other times, we want to be able to describe the length of an object for its own sake, not just to compare it to another object. Using units helps us to do this. We can do this using standard or non-standard units. Standard units are units that have been standardized (i.e., are always the same) and commonly understood, such as metric linear measures (centimetres, metres) whereas non-standard units are not. We focus on non-standard units in grade 1. These units allow students to explore the key ideas of measurement in a more concrete way than standard units can allow.
Non-standard units can be further broken down into uniform and non-uniform. Uniform units are consistent. Common examples include standard paper clips, interlocking cubes, or other standardized math manipulatives in your classroom. Non-uniform units are not consistent, such as: shoe length, hand width, arm span, pencils, and erasers. The advantage of these units is that they provide more concrete learning opportunities. Non-uniform units can be repeatedly used in an iterative process to measure length. For example, a student can walk toe-to-heal to measure the length of a room. You can do this with uniform units, too, though students can get more consistent results lining up multiple copies in a line. Key ideas to make explicit when measuring include: baselines (beginning to end), having no gaps or overlaps, measure in a straight line.
It is important to discuss how the unit size affects the quantity of the measure. For example, a classmate with smaller feet may measure that the classroom is 42 “feet” long whereas another student with larger feet may get 36 “feet” as their answer. The concept here is that the smaller the unit, the more of them we need to measure the same distance. This idea extends to all types of units and all kinds of measurements. Here are some examples to explore: Tiling an area with small squares requires more squares than using larger squares. You need more scoops to fill a bowl if your scooper is small vs large. You need fewer weights to balance a balance scale when weighing an object if the unit weight is larger.
Measurement and Geometry Foundations:
Foundational, supporting concepts and related competencies that are needed to develop this grade level concept:
- The concept of more, less, and the same measure using a baseline and direct comparison.
- Using comparative language, such as longer, shorter, taller, wider, heavier, lighter, holds more/less, etc. Adding the word “than” when explicitly comparing two items.
- Estimating and visualizing when comparing objects
Progression:
One way to progress through the concepts of measurement…
- Review direct linear comparison of objects (estimating first, then compare to a baseline, use comparative language to describe)
- Explore measuring objects directly using non-standard units
- Lining up multiple uniform units (e.g., manipulatives)
- Iterating non-uniform units (e.g., body measures)
- Formalizing key ideas in measurement
- Use of a baseline (i.e., starting and end points)
- Measure without gaps and overlaps
- Measure in a straight line
- Comparing quantity of unit relative to unit size
- Plenty of ongoing opportunities to measure objects (inside and outside of the classroom) using non-standard units and consolidating the key ideas listed above
Sample Week at a Glance:
Previous practice with a Number Line routine, such as the Clothesline Math routine, would allow students to make some useful connections between the Number and Measurement math strands. This prior knowledge is reflected in the sample lesson below.
Topic: Review of direct linear comparison of objects
Before: Gather students together on the carpet. Pull out two objects from the mystery bag. Ask students how they can tell which object is longer? The idea here is that students can compare the objects side by side to see which is longer. Emphasize the importance of lining them up using a baseline. Repeat this process with a few more objects, including two objects that are the same size. Discuss comparative language, such as: longer than, shorter than, the same length as.
During: Choose an item pulled from the bag (e.g., a sharpened pencil). Students are asked to look around the room in search of another object that is as close in length (height/width) to the object as possible. Students have a minute to find the object and bring it back to the carpet. Then ask volunteers to use direct comparison to see if the item they found is longer than, shorter than, or the same length as the teacher’s item. Discuss the use of a baseline, estimating lengths, and using comparative language. Repeat with a few more objects from the mystery bag.
After: Ask students to compare their items with another student’s items. Which items are longer or shorter? Are any the same length? How do they know? Have some students share their findings as a class.
Alternatively, have students pick two items they compared and draw/trace them in their journals to show how you can compare length using a baseline.
Topic: Explore measuring with uniform non-standard units
Before: Teacher prepares a number of bins of objects that can be used to measure items using non-standard units, such as linking cubes, colour square tiles, paper clips, counters, etc. The teacher shows students the picture of an item to determine the length of. A copy of this image is on a sheet of paper for students to use. Teacher asks students: How can we describe how long the image is using the materials in the bins?
During: Students explore this question in pairs using the various materials. Partway through, the teacher gathers students at the carpet to discuss measuring strategies such as lining up multiple copies of one kind of material in a line or using repeated units of a unit. The teacher emphasizes key concepts described in the overview (baseline, no gaps/overlaps, measure in a straight line). Then students continue exploring using the materials and recording their results.
After: Students gather at the carpet to consolidate the key concepts and measuring strategies. This time the teacher focuses on how different size units produce different answers, whereas students using the same unit should get the same answer. The teacher can use this image to help get the point across.
Topic: Body Measures Lesson, part I (non-uniform non-standard units)
Follow the three part lesson plan found here. This lesson needs a double block of time to complete. If doing over two days (as is described here), it is suggested to complete the “before” section on the first day and the “during” and “after” on day two. Some students may need additional time to complete their personal measuring tapes. Early finishers can use their measuring tapes to measure all sorts of things inside and outside of the classroom.
Topic: Body Measures Lesson, part II (making a personal measuring tape)
Follow the three part lesson plan found here. This lesson needs a double block of time to complete. If doing over two days (as is described here), it is suggested to complete the “before” section on the first day and the “during” and “after” on day two. Some students may need additional time to complete their personal measuring tapes. Early finishers can use their measuring tapes to measure all sorts of things inside and outside of the classroom.
Topic: Practice with measurement
Before: Number Line routine. Draw a number line on the board using a metre stick marking 0 (at the 0 cm mark) and 20 (at the 100 cm mark). Then ask students to estimate where they would place the numbers 9, 6, and 17. Alternatively, you can use 0 to 10 and the number 7 if this is too challenging. Discuss strategies, such as using benchmarks (5, 10, 15).
Connect to the previous lessons by showing how a measuring tape or ruler is a number line that can be used to measure objects. You can place items on the board with the baseline of 0 to measure them to make this point clear.
During: Math Workshop. Some ideas include…
- Meet with the teacher to formatively assess key measurement concepts: baseline, gaps/overlaps, measuring in a straight line, uniform vs non-uniform non-standard units, standard vs non-standard units (if students are curious). Have students measure various items while you observe, support, and provide individualized feedback as necessary.
- Online measurement games such as this one (K) or this one (gr. 1) or a Mathletics assignment.
- Measuring objects using uniform non-standard units. Provide baskets of items for units as well as objects to measure. This encourages students to line up repeated units.
- Measuring objects using uniform non-standard units. Provide objects to measure and single units to iterate. Alternatively, you can combine this station with the previous one and have students compare their answers using repeated units vs iterating.
- Measuring objects using non-uniform non-standard units. Provide objects to measure and encourage students to use body measures or their personal measuring tapes from the Body Measures activity.
Alternatively, some students may need time to finish their personal measuring tapes from the previous lesson.
After: Students and teachers meet to debrief the stations with a particular focus on any areas that students found challenging or interesting. For example, how did measuring an item with multiple copies of the same unit compare with iterating one of these units? How did using small units compare to using larger units? How can we be very precise with our measurements?
The following week could begin with outdoor opportunities for students to use their personal measuring tapes and some assessment of measurement concepts. Students can then explore tiling an area… How does the size of the square affect the measure of the area? How can we compare the area of two different rectangles? Why squares and not circles? (Think no gaps/overlaps.)
Suggestions for Assessment
What to look for:
- Uses a baseline to measure, and measures in a straight line with no gaps or overlaps
- Can measure using multiple copies of the same unit or iterate a single copy of a unit
- Personal measuring tape has consistent increments and is used effectively
- Understands the difference between uniform and non-uniform units
By the end of Grade 5 students will be able to:
- Reasonably estimate the length of an object using non-standard units
- Effectively measure a variety of objects using uniform and non-uniform units, whether it be with multiple copies of a single unit or iterating a single copy of a unit.
- Explain how the size of a unit will affect the measure of an object being measured
Suggested Links
- Clothesline Math routine or Number lines from High Yield Routines (NCTM)
- Math Workshop (Jennifer Lempp)
- Making Math Meaningful to Canadian Students, K-8 (Marian Small)
- Math in a Cultural Context > Big John and Little Henry by Seth Myers
Key Measurement and Geometry Concept 2: Comparison of 2D Shapes and 3D objects
Overview
As with any new topic, we want to keep Geometry as concrete and visual as possible. Prior to grade 1, students are exploring single attributes of 2D shapes and 3D objects (formerly called 3D solids) by playing with materials and sorting them. A free sort where students decide how they want to sort their materials is a great way to explore materials and a worthwhile way to begin grade 1 geometry. There needs to be enough variety as well as repetition to make the sorting practical and a learning experience, too. Students learn to recognize attributes such as shape, colour, size, number of sides, number of corners, lines vs curves, and 2D vs 3D. In grade 1, we want students to be able to describe how they sorted their shapes. The teacher may ask, “What was your sorting rule?” The teacher may then present the student with a new shape and ask them to describe how it is sorted according to their sorting rule and why. Teachers should also ask students to do directed sorts, such as sorting by a particular attribute (e.g., number of corners). By grade two, students are sorting with two attributes using tools such as a Venn diagram and describing how they placed their materials.
Students begin to learn the names of shapes and their relative positions using positional language (beside, on top of, under, in front of, up, down, in out, etc.) By grade 2, students should know the names of basic shapes, such as circle, triangle, square, and rectangle, but it is important to start using these names in grade 1 so that students have plenty of time to learn them. 2D shapes should be explored both in the classroom and in the environment. The focus in grade 1 is on comparing and contrasting the attributes among 2D shapes and 3D objects. A simple way to do this is to ask how two shapes are the same and how they are different. We also want students to describe how combining shapes can create new shapes or how a shape can be decomposed into two or more shapes (e.g., two triangles can make a rectangle). The same is true for 3D objects, though students do not have to know the names of 3D objects until grade 3. Again, you can start using these names in grade 1, but students should not be assessed on knowing the names at this grade level.
The importance of geometry goes beyond knowing the names of shapes. Students are composing and decomposing shapes, just as they do for numbers, to create new shapes. Shapes are the basis of many patterns, the area of rectangles connects to multiplication concepts (i.e., the array), and comparing and contrasting attributes teaches students to look for what is the same and what is different. This is a useful skill in concept attainment, generalizing patterns, and determining algebraic equations.
Measurement and Geometry Foundations:
Foundational, supporting concepts and related competencies that are needed to develop this grade level concept:
- Exploring and playing with a variety of materials and sorting them in different ways
- Counting to 10 (one-to-one correspondence)
- Practice with explaining rules
- Visualizing combining shapes with the aid of concrete materials
- Familiarity with technical language (e.g., names of attributes, shapes, positional language)
Progression:
One way to progress through comparing 2D shapes and 3D objects is to…
- Begin with free sorts to explore attributes and math terminology
- Students decide how to sort their shapes
- What was your sorting rule? Can you sort another way?
- Have students do directed sorts with a single attribute
- Explaining the sorting rule
- Justifying how to sort a new shape according to the sorting rule
- Use side-by-side comparisons of shapes
- How are they the same? How are they different?
- Compose shapes from other shapes
- What shape do you get when you combine two shapes?
- How can you make a shape by combining two other shapes?
- Find and describe shapes in the environment
- Use positional language to describe relative positions of shapes
- Describe shapes composed of other shapes
- Which shapes are most common? Which are not? Why?
Sample Week at a Glance
This week’s lessons are the starting point for comparing 2D shapes and 3D objects. Prior to this week, students were given experience with sorting various objects (buttons, attribute blocks, etc.) before sorting shapes in this week’s lessons. It is useful to have introduced instructional routines, such as Which One Doesn’t Belong? and Same But Different, as well as the structure of Math Workshop, prior to this week’s lessons, as well.
Topic: Free sort
Before: Students are put into groups of 3 or 4 students and given a large variety of shapes to sort. One way to do this is to provide each group with these shapes which they will have to first cut out (you do not have to use all of the shapes). Tell students that they will be sorting these shapes any way they would like and to be prepared to share their sorting rule.
During: Students are sorting their shapes in a collaborative manner. Visit each group and ask them about their sorting rule, why particular shapes are placed in certain groups, and how they resolved any disagreements. Provide them with a shape that hasn’t yet been sorted to witness their thought process as they tell you how they would sort it. You may want to take pictures of the sorting for each group for future class discussions. Early finishers can be asked about different ways they could have sorted their shapes.
After: Have students do a gallery walk to see how other groups sorted their shapes. Then discuss the sorts as a class. Some questions to consider are:
- How did you sort your shapes?
- Which shapes were most challenging to sort and why?
- Were there any disagreements over how to sort a shape?
- Were there misfits that did not belong to any group? Why?
- How would you sort your shapes next time?
Topic: Directed sort
Before: Which One Doesn’t Belong? Use this image. Encourage students to notice the different attributes that the shapes have. Which shapes are the same? Which one is different? Can they make a case for each shape?
During: Students work in pairs. Provide them with a subset of the shapes they were working with yesterday. Alternatively, use physical shapes such as 3D solids. Ask them to sort the shapes in the following ways (one at a time). You may need to explain the terminology used, though students do not need to memorize terms at this grade level.
- Number of edges/sides
- Number of faces (for 3D objects)
- Number of corners/vertices
- Curved vs not curved
Circulate while students work, observing how they sort and offering support where needed. Ask students how a particular shape is sorted and why? Are there any misfits or challenging shapes? Why?
After: Gather students as a class and ask them to share about the sorting activity.
- What was your strategy for sorting shapes?
- Which directed sort was easiest? Which was most challenging? Why?
- Which shapes were most challenging to sort and why?
- Were there any disagreements over how to sort a shape?
- Were there misfits that did not belong to any group? Why?
- Were you able to sort a shape in more than one way? Why?
Topic: Comparing and contrasting shapes
Before: Same But Different. Use this image. How are these images the same? How are they different? Help students to see that a cube is the 3D version of the 2D square and that its faces are squares.
During: Math Workshop. Some ideas you could include…
- Meet with the teacher who can show students the pictures of Monday’s student free sorts to ask students if there are any shapes that were not sorted in the correct group and why. Alternatively, the teacher can show students a sort that the teacher created for the purpose of critiquing (which would include errors or evoke some debate).
- Free sort with some new and old materials (e.g., buttons, attribute blocks, paper clips, beads, mixed beans, etc.)
- Sorting 3D geometric solids where students pick one of the sorting rules:
- Number of faces
- Number of corners
- Number of edges
- Online same and different game such as this.
- Use a small subset of the shapes from Monday. Have students sort them and glue them onto a page in their math journal with their sorting rule. You may need to provide printed sorting rules to glue into their journals as well. This works well as an “After” activity as well.
After: Students and teachers meet to debrief the activities. For example, you might ask which materials were easier, more challenging, or more fun to sort? Which 3D sort was the most popular: by faces, corners, or edges? Why? What did you learn from these activities? What do you still want to learn?
Alternatively, refer to the last bullet in the list above.
Topic: Composing and decomposing shapes
Before: Quick Images routine. Show students the image below for 1 to 2 seconds and then hide it! You may need to do this again once or twice more. The idea behind hiding it is to force students to use a strategy for remembering the image, such as one involving composing and decomposing shapes.
Discuss strategies for recalling the image. How did students “see” the image? What shapes did they see? Can they see it more than one way?
During: Tangram puzzles. Use physical tangram pieces or an online program such as Mathigon tangrams if physical materials are not available. Students rearrange and compose shapes to make a variety of images thereby solving the puzzles. There are many printable templates online (free and paid versions). Here and here are two examples.
Alternatively, students can solve puzzles made for pattern blocks rather than tangrams, since pattern blocks may be easier to come by. You can also use these on another day for further practice with composing shapes.
For students needing more support, provide puzzles that show the individual shapes rather than the silhouette of the shapes. You can further challenge students by asking them to create a particular image (e.g., dog) without a template.
After: Have students share their strategies for solving the puzzles. Which puzzles were easier versus harder and why? Were they able to solve the puzzles in more than one way? Why or why not? What did they learn about shapes through this activity? The big idea here is that we can compose shapes from other shapes as well as decompose shapes into other shapes.
Topic: Shapes scavenger hunt
Before: Classroom shape hunt. Ask students what 2D shapes and 3D objects they see in and around the classroom. Do they see shapes composed from other shapes? How would they describe these shapes? Although students are not expected to know the names of all the different shapes and solids, introducing this language is a good idea.
One way to do the shape hunt is to play “I spy”. For example, say “I spy with my little eye a shape that is made of two triangles” and students can either point to it or go stand next to it. Repeat this with a variety of single shapes and composed shapes (2D and 3D). You can show images instead of using terms to make this activity more accessible to students.
During: Outdoor shape hunt. This can be done two ways…
- Students can use a math journal to draw a variety of different shapes that they see outside on the school grounds, including shapes composed of shapes.
- Students can be given a worksheet with particular shapes to find, such as: triangle, square, circle, rectangle, cube, prism, cone, pyramid. Using sample images makes this activity more accessible. Using words is also a good idea to help expose students to math terminology.
Ask students to take a mental note of where they found these shapes or to record this in their journal or on their worksheet if they are able to. This is an opportunity for students to use positional language.
After: Meet with students as a whole class. Ask them what examples of shapes they found outside. Can they use positional language to describe where they found the shapes? (e.g., above, below, up, down, etc.) Which shapes could be composed of other shapes? How do they know? Can any of these shapes be decomposed into more than one shape? How do they know?
Students need ongoing practice with all math concepts. One week is not enough even if the major ideas of the concept have been discussed in full. Thus, in the weeks and months to follow, make sure to revisit previously taught concepts. Routines such as the ones listed in the Suggested Links below are a great way to do this. Ongoing assessment is also important (see below).
Suggestions for Assessment
What to look for:
- Consistently sorts shapes according to a single attribute and can explain the sorting rule
- Compares and contrasts shapes; that is, student is able to confidently describe which attributes are the same and which are different
- Composes shapes from other shapes and decompose a shape into two or more shapes
- Identifies shapes in the environment and can describe their location using positional language
- Is able to name common shapes (e.g., square, circle, triangle, rectangle)
By the end of Grade 5 students will be able to:
- Compare two shapes according to a single attribute using a sorting rule
- Make shapes by combining other shapes (e.g., make a square using two triangles)
Suggested Links and Resources
- Which One Doesn’t Belong?
- Same But Different
- Mathigon tangrams
- High-Yield Routines: K-8 (NCTM) – source of Quick Image above
- Math Workshop (Jennifer Lempp)
- Making Math Meaningful to Canadian Students, K-8 (Marian Small)