Beginning With “Because”

Recently we were discussing online1 how to help Geometry students be successful writing proofs. This is a perennial problem, because the logical thinking involved in deductive proofs is new and challenging to many high school students. I’ve found that verbalizing “Because statements” throughout secondary math helps lay the foundation for students to justify their mathematical ideas. 

What is a “Because Statement”?

A “Because statement” provides the reason for doing any of the mathematical operations or actions that we do. It can be written, spoken aloud, or thought about silently by students. Getting into the habit of thinking “I did this <math operation> BECAUSE <this is the reason>” helps students develop logical thinking and (more importantly) get the problem correct!

Pre-Algebra and Algebra 1

As students begin learning to solve simple equations, I rely on the big idea of “Inverse Operations” as a framework for our work.  Inverse operations are operations that undo each other, and students readily identify Addition & Subtraction and Multiplication & Division as two inverse operation pairs.  This framework enables success with one-step equations, but also carries through all of high school mathematics2, so we include new pairs in our inventory when they are encountered in later math classes (squaring & square rooting, logs & exponents, differentiation & integration, etc.).

For example, to solve X + 4 = 10, students identify the operation “happening to X” (adding 4), and then undo it (subtract 4 from both sides). We verbalize the because statement and annotate our work:

Solving X+4 = 10.
Subtract 4 from both sides "because subtraction undoes addition" 
Answer: X = 6

Note that I avoid non-mathematical language, such as “move 4 to the other side” or “cancel the 4” since those statements aren’t rooted in the math of the situation.  Also students can abbreviate on their papers “BC +/– inv ops” since it is the naming of the reason that is important; a full sentence justification isn’t necessary.

We can use the Because statements for more complicated equations, which is good for class discussion when there is more than one way to solve an equation:

Inverse operations works nicely for proportions, and avoids the ambiguous “cross-multiply” or the despised “butterfly trick”. Here, students identify the operation on X (being divided by 3) then multiply by 3 because that undoes division.

Identify that fraction X/3 means X divided by 3.
Solve X/3=4/7:
3(X/3) = 3(4/7)
Answer: X=12/7

Note that this works for much more complicated proportions and rational equations; since every numerator is being divided by their denominators, undo those divisions by multiplying everything by each denominator.

As Algebra 1 students become more skilled at solving equations, they don’t need to write the Because statements, just think of them as they are solving.

Geometry

Early in Geometry classes, students encounter the ideas of midpoints, bisecting, and segment addition.  Each of these problems is easy to solve, but it is worth highlighting the Because statement so students get used to giving reasons for their geometry work.  

3 questions about segments.
Left: AB=12, BC=5
If AD=20, what is CD?
Answer: CD=3 because AB+BC+CD=AD

Middle: AC=12, B is midpoint of AC
What is AB?
Answer: AB=6 because 1/2(AC)=AB

Right: AB=12, B is midpoint of AC
What is AC?
Answer: AC=24 because 2(AB)=AC

Another familiar problem type in Geometry is the “Angle Chase” problem. Speaking, thinking, or writing a Because statement enables students to solve correctly, and communicate their mathematical thinking to the teacher.

Angle Chase problem: 
Find angle DEF if triangle DEG contains triangle DFG so F is on EG.
Segments EF=FD=DG
angle FDG=50
Angle Chase problem with an equilateral triangle adjacent to two isosceles triangles.

Notice that the notation and language used in the Because statement can be as relaxed or rigorous as needed in your class (here I didn’t use “m<DFE” notation, or distinguish between equal lengths of segments and congruent segments).

A common goal in many Geometry classes is to use geometry diagrams to review algebra skills (so students are prepared for Algebra 2, in theory). Whenever my students write an algebraic equation to solve, I ask them to write a Because statement based on the geometry situation. Take a look at each of these diagrams below, and think about the Because statements your students might write for each.

3 geometry problems with algebra expressions

The work that students do with Because statements is very helpful when they encounter proofs (two-column or otherwise3) in deciding which reason to use for a particular step. In the proof setup #1 (left below), students easily write the given statements. However, when looking for a reason that segment VX is congruent to XY, they sometimes want to use “bisector” in the reason. Instead, a Because statement must focus on properties already stated, so students correctly use “because X is the midpoint” instead of saying something about bisection.

2 Geometry Proof situations for proving congruent triangles.

In the proof setup #2 (right above), I deliberately present the givens without identifying what we need to prove. Students who have practiced Because statements are able to decide which triangles might be proven congruent based on the available information.

Algebra 2 and Beyond

Here are a few instances where students can use Because statements in more advanced courses:

  • End behavior: I know the graph of the polynomial will be “down-down” because the degree is even and it has a negative leading term.
  • The graph of this rational expression has a vertical asymptote at X=2 BC there’s a factor of (X–2) in denominator. [OR X=2 makes the denominator = 0]
  • To solve: (x-3)^\frac{3}{2} =8, raise both sides to 2/3 power because it undoes 3/2 power. [OR the power of 3/2 means cubing and square root, so need power of 2/3—squaring and cube root—to undo it]
  • I know the graph is increasing because dy/dx is positive.
  • I set y´´ = 0 because looking for inflection point.

Benefits of Because Statements

When students get in the habit of articulating Because statements, they are making sense of the mathematical steps they take and improving their reasoning skills. This helps support their understanding of proofs, for Geometry and beyond.


Notes & Resources:

1The online math educator community uses the hashtag MTBoS (an acronym for “Math Teacher Blog-o-Sphere”) and iTeachMath on a variety of platforms (Twitter, Facebook, Mastodon, etc.). If you are looking for helpful educators, shared resources, and thoughtful discussions, find us wherever you are online.

2Inverse Operations is a rule that “doesn’t expire” because it continues to hold true throughout later mathematics. Rules that expire should be avoided; to read more on this, see this series of articles from NCTM.

3Flow Proofs are a great way for students to visualize how statements are related in a proof, since they can see the “flow” of ideas instead of a list. Here are some examples and explanations.

This post was included in the Carnival of Mathematics 211, a monthly roundup of math blog posts from The Aperiodical.


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Reflections and Tangents by Karen D. Campe is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
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