Teaching Routine for Building Important Mental Math Strategies

Number partners are pairs of numbers that add to a third number. Given the landmark quality of powers of ten in our counting system, partners of powers of ten(1.0, 10, 100, 1000, etc) are immensely useful tools. Mastery of partners builds speed, accuracy, and most importantly, understanding place value and how numbers work and fit together. That foundational understanding is a gift that will keep giving through a student’s entire mathematical career.

Note that I did not say ‘memorization’ of partners. Though the process of mastering partners will by natural consequence lead to memorization through frequency of use, skipping to the memorization part voids all of the advantages understanding brings. A student who can intuit why 1000 is 100 x 10 is a student who will master 1000 x 0.1 as naturally as taking the next step, while a memorizer will need their hand held by an algorithm. This hand-holding is of no help recognizing if their answer even makes sense.

Consider this example of partners of 100 coming in handy. Suppose you need to find 99 times 27. You know 100 x 27 is 2700, but you only need 99 of the 27s. So, you need 2700 - 27. Since you have been thinking about partners of 100, you think 27 + __ = 100, that’s got to be 27 + 73 = 100. In between 2600 and 2700 is that 100. Therefore 2700 - 27 is 2673.

Fortunately, there is an amazing teaching routine for building partner sense that is accessible to any teacher and any student that can add single digit numbers. That routine is “I Have, You Need”. The teacher (parent) says, “For a total of 100, I have 46, you need?” The student responds with the partner. It can be fun to switch roles!

I covered this routine in depth in this podcast episode. Today I’m going over other number partners outside powers of ten that are almost as useful. In particular, partners of 30 and 60 are useful in multiple contexts. Most obviously is their use in any math involving time. Quick and accurate evaluation with these partners equals quick and accurate calculations with how much time is left to complete a task, or what time it will be when you finish. Time management can already be stressful enough without dealing with math fright at the same time.

In addition, partners of 30 and 60 are very useful for building fractional reasoning. 30 and 60 are easily divisible by 3 when compared to whole powers of 10, but are still also easily divisible by 10, 5, and 2.

For example, when finding ⅓ + ¼, one can think about hours and clocks. 

Since ⅓ of an hour is 20 minutes and ¼ of an hour is 45 minutes, ⅓ + ¼ = 20/60 + 15/60 = 35/60. One can then reason that 35 minutes in on the 7 on a clock, 7 out of the 12 5-minute chunks, 7/12. You could have actually started there, ⅓ of an hour is on the 4, ¼ of an hour is on the 3, so ⅓ + ¼ = 4/12 + 3/12 = 7/12. Reasoning about denominators that are fractions of 60 is so slick using time and a clock to think about equivalent fractions.

Check out all the fun Kim and I had messing with this new way to play I Have, You Need here.