Examples:
TABLE2.10:
Cross Multiply Don’t Cross Multiply
3
4 =10
x3
4 +10
x+ 3 x
4 =10
xx+ 3
4 −10
xOnly Cancel Common Factors –When reducing a fraction or putting a ratio in simplest terms, students often try to
cancel over an addition or subtraction sign. This problem occurs most frequently when students work with fractions
that contain variable expressions. To combat this error, go back to numerical examples. Students will see that what
they are doing does not make sense when the variables are removed. Then go back to example with variables.
Hopefully the students will be able to carry over the concept.
Examples:
TABLE2.11:
Can be Reduced Can’t be Reduced
3 · 2
5 · 23 + 2
5 + 2
3 (x− 4 )
3 · 2x− 4
4Color-Code the Proofs –The proof in this section requires many substitutions of similar looking expressions. It is
difficult to see where everything is coming from and moving to. When presenting the proof in class use colors so the
variables will be easier to follow. Another option is to have the students do the color-coding. Once they understand
the mechanics of the proof in the lesson, they will be able to do the similar proof in the exercises.
Similar Polygons
A Common Vocabulary Error –Students frequently interchange the words proportional and similar. Remind them
that proportional describes a relationship between numbers, and similar describes a relationship between figures,
like equal and congruent.
Compare and Contrast Similar with Congruent –If your students have already learned about congruent figures,
now would be a good time to review. The definitions of congruent and similar are very close. Ask the students if
they can identity the difference; it’s only one word. You can also point out that congruent is a subset of similar like
square is a subset of rectangle, or mother is a subset of women. Understanding the differences between congruent
and similar will be important in upcoming lessons when proving triangles similar.
Use that Similarity Statement –In some figures, which sides of similar polygons correspond is obvious, but when
the polygons are almost congruent, or oriented differently, the figure can be misleading. Students usually begin by
using the figure and then forget to use the similarity statement when necessary. Remind them about this information
as they start working on more complicated problems. The similarity statement is particularly useful for students that
have a hard time with visual-spatial processing.
Who’s in the Numerator –When writing a proportion students sometimes carelessly switch which polygon’s
measurements are in the numerator. To combat this I tell the students to choose right from the beginning and BE
CONNSISTENT throughout the problem. When it comes to writing proportions if the students focus on being
orderly and consistent, they will usually come up with a correct setup.
Bigger or Smaller –After completing a problem it is always a good idea to take a minute to decide if the answer
makes sense. This is hard to get students to do. When using a scale factor, a good way to check that the correct ratio
2.7. Similarity