Similarity: Vocabulary Rating, Mental Math

Two strategies and a lesson plan from the course I took on Differentiation in Math:

Strategy: Vocabulary Rating Chart

Geometry is a very vocabulary heavy course.  For each unit we introduce new words and continue to apply previously learned words.  In the similarity unit we compare congruency to similarity, introduce the word dilate, use the words ratio and proportion for the first time since Algebra and introduce the geometric mean.  I find that students are familiar with most of the vocabulary words we use but often have misconceptions either related to poorly learned math definitions or difficulty bridging general use definitions with the more precise mathematical ones.  It would be helpful to see what knowledge students are coming in with to address those misconceptions right away.  Having students complete this chart at both the beginning and end of the unit would allow students to see what they have learned and highlight what they need to review before an exam.  

Strategy: Mental Math String

This mental math string starts with a polygon (review from the previous unit), then uses fraction and similarity vocabulary.  Square roots come up during this unit as we study geometric means.  This activity would be great to do once as a whole class, then repeat in small groups where students would make up their own string for their peers to try.  Having students write their own string would encourage them to use appropriate vocabulary (I would suggest they look at their vocabulary knowledge rating chart) and gives them the opportunity to be creative while challenging their peers in a low-risk competition.

• Start with the number of sides in a hexagon.

• Add the numerator of 4/7.

• Scale by 5.

• Multiply by the denominator of 9/2.

• Take the square root.

• Scale by 1/5

Was your answer 2?

Learning Activity: Introduction to Similarity

Grade Level: 10

Topic: Similarity

CCSS: G.SRT.2

Learning Objective: Students will discover the characteristics of similar figures.

Description:

As students enter class hand them the Vocabulary Rating Chart.  Encourage students to guess at definitions and think about whether they have heard the words both in and out of math class.  Do not define any of the words yet as we will be seeing the words again throughout the unit and by the end of our study of similarity students will have a precise definition of each word.  Distribute Similarity Cards and ask students to work in pairs sorting the cards by any characteristics they would like.  Move around the room encouraging students to explain their categories and try multiple ways of sorting the cards.  Ask students to share their approaches by sorting the cards on the SMART board and defining each category.  Save a group who sorted by similar figures for last.  Acknowledge these are called similar figures and share the mathematical definition of similar (the transformation definition).  Have students sort their cards to match the ones on the board.  Using rulers and protractors students should work together to measure the sides and angles of all the figures.  Have them record observations and discuss characteristics of similar figures.  By the end of class make sure everyone has determined that corresponding angles are congruent and corresponding sides are proportional.

Resources: 

• Vocabulary Rating Chart (see above)

• Similarity Cards (Cards depicting a variety of shapes where each shape has at least one similar pair.  Be sure to include triangles/rectangles/etc. that are not similar to show that not all shapes with the same number of sides are similar.)

• Rulers, protractors

Assessment: 

While students are sorting cards, collect and flip through the Vocabulary Rating Charts; note any student who has an inaccurate definition of congruent or corresponding and check in with them (use the vocabulary with them while discussing the categories they created).  If the class as a whole seems uncertain on ratio and proportion (again gather evidence from the Vocabulary Rating Charts) take a few minutes to review that topic part way through the measurement process (after they have started to see the relationship of the sides but before they try to write a generalization).  If students are struggling to see a relationship between the sides during the measurement activity, point them toward a set of shapes where the scale factor is two, most students recognize doubling faster than any other ratio.  If students have already found the relationship between the sides or angles, challenge them to predict the next value before measuring.  Students who finish early could be asked to verify their angle measurements by comparing to the known sum for each polygon (an opportunity to review the angle sum rules).  They could also be asked to find the precise scale factor for each set of similar figures.  At the beginning of the next class give a quiz.  Sample questions: “Define the term similar.” “Given these figures, determine if they are similar and explain your reasoning.”   At the end of the unit give the Vocabulary Rating Chart again.  

Analysis of Differentiation: 

In this lesson the content is differentiated when the teacher gives additional instruction to either individual students or the whole class based on the results of the Vocabulary Rating Charts.  The content and product is differentiated when advanced students are asked to complete an additional analysis which includes review material (angle sum rule) and extension material (scale factor).