Math Matters

The Importance of Spatial Reasoning

What is Spatial Reasoning?

Spatial reasoning is defined as the ability to imagine things in three-dimensions. It includes the ability to move objects around in your mind. Spatial reasoning gives your students a means to visualize the math in their heads and connects to so many concepts students need to be strong mathematicians.

Spatial abilities include being able to think about:

how objects look when rotated (e.g., if you turn a V to the side, where will the opening be)

how objects look from different angles, (e.g., what a pyramid looks like from the side or if you were looking at it from above)
how objects look on the inside (e.g., if you slice a cylinder down the middle of the circle on top and open it up you will see a rectangle!)
how parts of an object fit together, such as being able to imagine that if two triangles are put together a certain way, they would make a rectangle
how positions of objects relate to each other, such as understanding that a car is inside a garage or a tree is behind a fence.

Information taken from https://dreme.stanford.edu/news/spatial-reasoning-why-math-talk-about-more-numbers; Development & Research in Early Math Education; Spatial Reasoning: Why Math Talk is About More Than Numbers.

Click each picture for a suggestion.

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Visualize the shapes found within another figure, such as this question.

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If you compose a cube out of other cubes, and paint the surface, how many cubes have paint on 1 side, 2 sides, no sides?

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Use pattern blocks or tangram puzzles to compose or decompose figures. What is the fewest blocks needed to fill the figure? What is the greatest number of blocks needed? (a pattern block template is available for printing your own on my website downloads page if you don't have blocks at home.)

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What skills does spatial reasoning develop?

Many math concepts develop from a student's experiences with spatial reasoning including:

· Location, distance, direction

· 2D and 3D shapes and their attributes

· Patterns, that lead to algebraic relationships

· Symmetry and geometric transformation

· Geometric measurement

Spatial reasoning skills support the arts and artistic careers, along with many career fields in STEM that now use 3D technology such as medicine, architecture, computer sciences, chemistry, geology, geography and more.

Spatial reasoning is important because we live in a 3D world!

The paper in the pic was folded in half horizontally, then vertically, then horizontally again. The left bottom corner was cut out. What will the paper look like when its unfolded?

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Use blocks to show a pattern. Extend the pattern to the next figure(s), make a 4 column table of figure number, drawing, process, and number of blocks. Use the table to develop a rule. Use the rule to project other values (How many blocks in the 50th figure? What figure number will have 39 blocks?)

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Use blocks to build 3D figures. Look at the views (top, bottom, sides, front/back). Find volume and surface area. Make a net of the figure. Draw another figure using only the views and have someone try to recreate the block figure.

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How many unique shapes can be made using a set number of blocks? Are the different orientations of the same thing equivalent? Do they count as different? Use this to explore ways to change orientation without changing congruence (transformations)

Like the visual memory needed to recreate an image, subitizing involves knowing a quantity when viewing an arrangement, such as using flash cards of dominos or playing cards w/o numbers and knowing how many are there.

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Activities you can do at home.

Composing/Decomposing shape patterns using blocks. I have a free template on my website you can print and cut apart if you don't have any shape blocks at home.
Spatial visualization - looking at a 3D figure from different perspectives, or imagining the hidden cubes.
Nets and folding activities - What does the net look like? How many ways can the net be made?
Congruence through Orientation Problems and Transformation - How many unique 3D figures can you build using a set number of cubes? Modeling transformations of a figure (translation, rotation, reflection on or off a plane)
Working visual memory - Draw a shape or figure. Hold it up for 5 seconds. Then have students try to recreate it from memory. Other skills in this area include subitizing* , number line relationships, and measuring with a ruler, especially from a point other than zero. (See info on subitizing below)
Patterning - build several figures in a pattern. Students can identify the pattern, extend it, or develop a table that leads to a rule.
Use graphs and maps - interpret patterns in functions or on plots, investigate maps, explore patterns in other types of graphs.
Model Pythagorean relationships, similarity vs congruence, explore the views of a figure, and investigate right triangle relationships including trigonometry.

Some information taken from "Paying Attention to Spatial Reasoning K-12: Support Document for Paying Attention to Mathematics Education" Ontario Ministry for Education

http://www.edu.gov.on.ca/eng/literacynumeracy/lnspayingattention.pdf

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Nets & Attributes of 3D Figures activity

If you could use an activity on nets, I have one! And for the COVID-19 distance learning time period, this activity is available in a student-friendly, shareable format free of charge. I even have a data table on google docs you can use to share with students at home for recording. They could print cards at home if they want, or just refer to the pdf document and record the information to share back with you as desired.

The activity instructions in printable/shareable pdf is linked here.

The data table link is found here. Be sure to save a copy to your own google drive and make sure your students do the same so they can't change the original doc.

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Develop skills by printing isometric dot paper to draw a representation of a 3D block-figure on the paper from one perspective, or all of them for more practice. Talk about hidden cubes in the drawing.

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Represent on grid paper the different views of a block-figure. Then trade the views paper with someone and try to recreate the figure on the grid paper using blocks.

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Make a figure and draw it on the coordinate plane. Explore transformations first using physical recreations of the shape, and then color them in. Do all congruence transformations, or one at a time. Extend this to dilations (similarity and scale drawings).

Dynamic Paper

Need grid papers, dot paper, etc? Try this link.

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What is Subitizing?

Subitizing is the ability to recognize quantity as a number without counting. When you look at the dice in the picture, you should be able to quickly, accurately, and confidently state the number shown on each. Children develop this skill in early grade instruction (K-1) in mathematics. This skill is part of visual working memory which is all part of spatial reasoning.

Use dominos, dice, playing cards without numbers, tally marks, ten frames, or even fingers to have students develop this skill.

As students get older, use money, 100 grid models, and images of number lines (zero to one marked) along with a point marked on the interval. Have students try to identify the value of a quantity as quickly, accurately, and confidently as possible and keep developing this valuable skill!