Summary
Proofs can be difficult and require a certain mindset. Spatial reasoning and awareness is a skill that can be developed no matter the age of the student. In this lesson, spatial reasoning is refined using drawing techniques and prior knowledge of shapes, lines, and the coordinate plane. This lesson is a great culmination activity for algebraic reasoning and proofs leading into geometric and figure proofs in a Geometry class. Required knowledge for this lesson includes: the ability to calculate midpoint, slope, distance, angle measure.
Essential Question(s)
How does spatial awareness help us to be confident in our geometric knowledge?
Snapshots
Engage
Students attempt a Quick Draw of a coordinate image and reflect on what they looked for first.
Explore
Students use calculations to refine their Quick Draw to be exact.
Explain
Students take notes over academic language related to proofs.
Extend
Students look at sample problems to determine what is given and what else they can figure out from the given.
Evaluate
Students reflect on what they 'look' for first and make a plan for tackling proofs in the future.
Materials
Lesson Slides (attached)
Evaluating Proofs handout (attached, 1 per student)
Metacognitive Geometry Plan handout (attached, 1 per student)
How I Know It handout (attached, 1 per student)
Students' interactive notebooks
Basic math tools (rulers, protractors, compass, etc.)
Engage
Begin the lesson by introducing the essential question and objective with students using slides 2 and 3 of the attached Lesson Slides.
Display slide 4. This slide has the directions for the Quick Draw Activity. While students are reading the directions, pass out a copy of the How I Know It handout, which contains a coordinate grid and a How I Know It circle, to each student.
Remind students that the image will be quick, so look carefully. Progress to slide 5—click once to trigger the image to appear. The image will appear and automatically disappear after three seconds. Once the image disappears, tell students to replicate as much as they can on their grid paper.
Go to slide 6. Invite students to use the second page of the How I Know It handout to answer two questions: 'What did I look for to help me?' (inside the circle) and 'How do I know this is something I should look for?' (outside the circle with a line connecting to what they looked for). Have students think back, using their drawing as inspiration, to identify what 'jumped out’ in the image that was easiest to replicate and how they knew they could rely on that piece of the image.
Let students know that they are going to see the image again, so they should make a plan of what to look for to help replicate the image perfectly. Go to slide 7 to display the image. Click once to trigger the image to appear. The image will appear and automatically disappear after three seconds once again.
Go to slide 8. Again, have students replicate the image from memory and reflect on what they looked for and how they knew to look for that.
Explore
Pass out rulers, protractors, and calculators. Progress to slide 9 to give students a few vital clues to create an exact replica of the image. Remind students that they can use any of their knowledge or formulas if they think it applies.
Give students about 10 minutes to make their calculations and finish their quick draw.
Have students look back on their How I Know It answers and denote which one they measured, calculated, or used a formula for.
Explain
Have students set their drawing aside and set up the next page in their notebook for the notes.
Progress to slide 10. Remind students that they will take notes over the information.
Explain that looking at an image and seeing it mathematically is sometimes believed to be a talent that “you either have or don’t”, but in reality, it's a skill that can be improved with practice. One of the ways to improve looking at things mathematically is to know what to look for, and understanding why.
Progress to slide 11. It lists the following terms with their definitions: undefined, defined, postulate, theorem, given.
Go to slide 12. Direct students to look back on the How I Know It answers. Using the information they just took notes over, tell students to classify all the things they looked for as either undefined, defined, postulate, or theorem.
Extend
Go to slide 13. Pass out an Evaluating Proofs handout to each student.
Tell students that they will not solve the problems, but they will use their new skills to observe what is there and what they think is the most important information they will use for the problem. Also, they need to think about what is given as well as what they would need to figure out.
Evaluate
Go to slide 14. Pass out the Metacognitive Geometry Plan handout. Explain to students that they are going to reflect on their learning during the lesson and make a 'plan' for how they are going to look at geometric figures throughout the school year.
When students are done with the handout, have them glue or attach their plan in their notebook for use later.
Resources
K20 Center. (n.d.). How I know it. Strategies. https://learn.k20center.ou.edu/strategy/144
K20 Center. (n.d.). Quick draw. Strategies. https://learn.k20center.ou.edu/strategy/51