Experience First

This lesson starts students off with some elementary school math and builds up concepts all the way through early Calculus! Questions 1-3 have students think about adding fractions with unlike denominators. The denominators are purposefully coprime so that students see that the denominators of the fraction break-downs are in fact the factors of the original denominator. Students will likely use strategic guess and check to find A and B, thinking about how many multiples of 6 and 7 (or 13 and 4) are needed to arrive at the numerator. Use the debrief to highlight the relationship between the numerators.

 

Question 4 highlights the Algebra 2 skill of adding rational functions.   

 

In question 6 we ask students to think about equivalence and like terms. Students should notice that linear terms on the left side of the equation must equal linear terms on the right side and the same with constant terms. Look for groups that can articulate this idea for 8a and have them share in the debrief.

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Formalize Later

In the debrief, we formalize students’ intuitive understanding of how to find the values of A and B. The process of decomposing fractions has students shift regularly between combining fractions by creating a like denominator and thinking about a fraction as its component parts. Emphasize the idea of factoring and making common denominators over procedural “tricks” like the criss-cross or butterfly method. Make sure students can articulate why we multiply by the opposite denominator.

 

Students will likely not use systems of equations to find their values of A and B, so this becomes an important add-on in the debrief, tying this lesson back to previous lessons and reinforcing ideas of equivalence. We don’t require that students write the system if they can “eye-ball” it, but students should be able to explain the idea of equating constant and linear terms.

 

Continue to use bridging language to help students master new vocabulary. (Ex: When adding rational functions-- functions that look like fractions--we need to…) When students use phrases like “the terms with x” or “the terms that are just numbers” use subtle revoicing to layer on the vocabulary of “linear terms” and “constant terms”. The goal is not necessarily to correct students outright, but to continually link informal student language with more formal mathematical language.

 

Note that we focus exclusively on linear, distinct factors. You may wish to extend this lesson to include more complicated partial fraction decompositions.