Last month, Jenna Laib (@JennaLaib) tweeted about whether there was a name for pairs of fractions that added up to 1, such as ¼ and ¾. And Susan Russo (@Dsrussosusan) wondered what to call numbers that add up to 100%.
That led me to think about all the fabulous number pairs we teach to our students; read on for these and other Powerful Pairs in math class!
1. Addend Pairs
Jenna’s original idea came from a common use in elementary classrooms: what pairs of numbers add up to 10? This is a key concept for students, and knowing the relationships 1+9, 2+8, 3+7, 4+6, 5+5 and all the reverse orderings is one of the content standards beginning in Kindergarten*. It is a foundational skill that so much more mathematics is built on.
Later, students explore the addend pairs that make 100, first with groups of 10 (e.g. 70+30) and later for any integer between 0 and 100. Other “landmark numbers” might be explored as well (some teachers call these “number bonds”).
Jenna didn’t immediately have a name for the pair of fractions whose sum = 1, so she used “complementary” with some students but asked the math twitter community (#MTBoS and #iTeachMath) what we thought. Suggestions included: unit pairs, unit partners, unit pair fractions (which are not necessarily unit fractions), and unit complement. Susan’s query for pairs like 85% and 15% that we might use in percent decrease problems, yielded “complementary percentages” and the quirky “cent-lement”.
I think “complement” is a good term to use here, even though it has some other mathematical meanings. In probability, two events are complementary if their probabilities add up to 1 (because one event only happens when the other does not); this is actually exactly what Susan was asking about.
2. Zero Pairs
Numbers that are opposites have the same absolute value but different signs, so their sum is zero, thus the name zero pairs. This may seem mundane, but this is one of the “inverse operations” that allows us to solve algebra problems such as X + 5 = 12 or X – 3 = 10. We “undo” the operation that is shown by creating a “zero pair” (and I prefer these terms instead of “move the number to the other side”).
Howie Hua (@Howie_Hua) uses the property that a number and its opposite add up to zero in arithmetic problems; check out his video here. You can use this property to make computations like 117 + 58 easier; just put in a zero pair of + 2 – 2 like this:
3. Factor Pairs
Factor pairs are pairs of numbers that multiply to equal a certain product number. We can model products with area tiles** and notice that there are 3 ways to factor the number 12, for example.
Prime numbers will have only 1 pair of factors, and square numbers will have one factor pair in which the numbers are the same (the square root).
Factor pairs are powerful tools for students to use when doing arithmetic and algebra. One of my go-to computation strategies is to decompose into known factor pairs and regroup conveniently (see below). In algebra, knowing factor pairs helps with factoring trinomials, especially if one of the coefficients is prime.
4. Conjugate Pairs
Conjugate pairs are numbers or binomial expressions that “play nicely together”†. They are of the form A + B and A – B, meaning that the first terms are the same and the second terms are opposites. They crop up in several places in high school mathematics.
Algebra 2 and PreCalculus students learn about complex conjugates a + bi and a – bi , with equal real parts and opposite imaginary parts. When multiplied, the product of complex conjugates is the real number a2 + b2 with the imaginary part eliminated.
Other conjugates work in a similar way. Conjugates involving a radical when multiplied will remove the radicals, which is helpful if you happen to be rationalizing a denominator.
I’ve started discussing conjugates with my Algebra 1 students so they have an easier time recognizing them‡. The conjugates (a + b)(a – b) are the basis for the “difference of two squares” binomial pattern a2 – b2. We called the conjugates the “add-subtract pattern” at first and noticed that multiplying conjugates yields two terms that cancel away (“add to zero”) so we can save ourselves some of the distributing steps we usually do (the O and the I in “FOIL” if you use that terminology).
The two solutions to a quadratic equation will be conjugates, which is easiest to see when using the quadratic formula to solve. The common use of ± instead of writing separate solutions can mask the idea (for some students) that the two solutions will be equal distances from the line of symmetry x = –b/2a.
5. Geometry Pairs
Our final set of powerful pairs are some wonderful angle pairs in Geometry. Here we use the term “complements” again, since Complementary Angles are a pair that add up to 90°. Complementary angles can be made by partitioning a right angle, but they also appear separately, such as the two acute angles in a right triangle (angles A and B in the triangle below).
Supplementary Angles are a pair that add up to 180°. If supplementary angles are adjacent (having a common side) then they are called Linear Pair Angles like these angles 1 and 2, making a Straight Angle. Linear Pair angles and Vertical Angles are two useful angle pairs that can be assumed from a diagram without other markings§, astute Geometry students know. Non-adjacent supplements pop up inside trapezoids and parallelograms (or anywhere parallel lines are cut by a transversal).
I wondered if there was a name for angles that add up to 360° since that is another useful pair that shows up when we study circles. Indeed, there is! They are called either Conjugate Angles or Explementary Angles¤: each one is the Explement of the other and create a Complete Angle. When two arcs of a circle have the same endpoints and don’t overlap (one is the major arc, the other is the minor arc), they are called Conjugate Arcs and add up to 360° because they create a complete circle. Minor arc IJ and major arc IKJ are conjugate arcs in the circle below.
What a list we’ve covered of pairs of numbers that add to 1, 10, 100%, 90°, 180°, 360° and more! Even if you don’t remember all the names for these powerful pairs, the important thing is to be able to use them productively for doing math!
Notes & Resources
*Addend pairs that equal 10 are a Kindergarten standard: CCSS-M-K.OA.4 For any number from 1 to 9, find the number that makes 10 when added to the given number.
**These factor pairs are modeled with number tiles available on the Mathigon Polypad at https://mathigon.org/polypad. Their game Factris is a great way to practice factor pairs visually, but be warned, it’s addictive!
† This phrasing came from Beth Hentges (@bethhentges) who noticed that the Spanish verb for playing is part of the word conjugate in this tweet.
‡ The difference of two squares pattern has always been part of the Algebra 1 curriculum, but the binomials that they factor into are rarely called conjugates at this level. I have been using the term both as a preview for later math classes, and also for its power to highlight the useful pattern.
§ Vertical angles and linear pair angles are two of the most useful angle pairs in geometry because these relationships hold true anytime lines intersect. When students are studying angles formed by parallel lines and a transversal, they need to learn several new angle relationships. I find it helpful to distinguish between the properties that are true even when the lines aren’t parallel, from the properties that are only true when the lines are parallel.
¤ These and other terms were defined in The Penguin Dictionary of Mathematics 2nd Edition, edited by David Nelson (Penguin: 1998). Another helpful source is The Words of Mathematics: An Etymological Dictionary of Mathematical Terms used in English by Steven Schwartzman (MAA: 1994).