Lines cut by a transversal is a pivotal concept in geometry. This post discusses the importance and gives a strategy for conceptual understanding as opposed to memorization.
Spatial awareness is an important part of geometry. Students with spatial awareness difficulties will struggle with even the simplest-seeming concepts. As teachers, we can sometimes lose patience when students don’t “get it” as quickly as feel like they should, or as quickly as the pacing guide suggests.
Spatial awareness is the ability to be aware of oneself in space. It is an organised knowledge of objects in relation to oneself in that given space. Spatial awareness also involves understanding the relationship of these objects when there is a change of position. It can therefore be said that the awareness of spatial relationships is the ability to see and understand two or more objects in relation to each other and to oneself. This is a complex cognitive skill that children need to develop at an early age. Spatial awareness does come naturally to most children but some children have difficulties with this skill and there are things that can be done to help improve spatial awareness.
In the classroom the child with spatial awareness difficulties often finds mathematics hard. This is due to the abstract concepts of the subject especially where shapes, areas, volume and space is involved. They will have problems reproducing patterns, sequences and shapes. Their strengths, however, are with the more practical and concrete subjects. These children will often find that they excel at using a multisensory way of learning. They tend to have good auditory memory skills and have strength in speaking confidently whilst being able to listen well. They tend to have good verbal comprehension skills and their strength is usually in verbal and non verbal reasoning.
One concept that challenges students’ spatial awareness early on is the relationships between angles created by lines but by a transversal.
The angle pairs created have cool relationships. When the transversal cuts through parallel lines, the alternate exterior angles, alternate interior angles, corresponding angles, and vertical angles are congruent while the same-side exterior and same-side interior angles are supplementary. Once students understand these cool relationships and that the relationships don’t hold true when the lines are not parallel, they can apply the knowledge to more challenging problems and in new ways. But, how do students access these more challenging problems of application when they don’t fully understand these basic relationships? The answer? They can’t.
So, how do we as teachers make this understandable for all students? Even the ones that have spatial awareness difficulties? Well, there are several different ways- none that work perfectly for every student. But the approach described below will work for most students. Then, through targeted small group/individual intervention and reteach you can support the few that need more scaffolding.
Here’s what you will need:
- painters’ tape/floor tape
- dry-erase markers
- some intentional questions.
Use space on your floor or your hallway to place the tape. You need at least two stations- one for parallel lines and one for non-parallel lines. You can have more than one of each if space allows. Be sure to number each angle for reference. The first class I tried this with struggled a lot with talking and writing about their observations. It didn’t take me long to realize this was ENTIRELY my fault and I had not set them up for success. I just numbered 1-8 on some post-it notes very quickly, but if you plan on leaving the lines on the floor beyond this activity (which I suggest doing), you may opt for something a bit more sturdy/permanent. Name each set of lines station A, B, C, etc. DON’T CALL THEM PARALLEL AND NON-PARALLEL! That will give it away.
Divide the students into groups- the same number as you have stations. Give each group some transparencies and dry-erase markers. Have them trace angles and compare/contrast the sizes of the angles. If your students struggle with this, here are some prompts/starters you can post or give to them.
- What do you notice about the lines?
- What do you notice about the angles?
- I notice ∠ ____ and ∠ ____ have the same measure.
- I notice ∠ ____ and ∠ ____ have different measures.
- Why do you think that is?
Have the groups rotate so that each group has investigated at least one parallel and one non-parallel station. Send groups back to their seats for table/group discussions. Then have a whole class discussion that leads them to the conclusion of the angle pair postulates and theorems. Lead them to the realization about parallel vs. non-parallel. Try not to give it away!
Once students have an understanding of what they have discovered, provide some structured notes to help them synthesize their thoughts. Click here to get my FREE guided notes for interactive notebooks.
Last of all- have fun with your students! Happy learning! If you are interested in reading about more ideas for Geometry instruction, try this post over introducing a new unit.