T Kaap

Math for Archers

Many parts of archery are a lifelong challenge, and that’s part of the glory of the game.  But arithmetic shouldn’t be one of those things.  So here are some thoughts and strategies specifically to help with doing archery math more confidently and more accurately.  Once you find a technique that works for you, then it’s the right technique.

First, Some Notes:

The world isn’t split up into “people who are good at math” and “people who aren’t good at math”.  This isn’t like “being tall” that can’t be changed.  Skill with arithmetic can be learned by practicing it.  Math is such an enormous field of study that it’s impossible to be “good” at all of it.  Being good at math doesn’t make you a better person.  Struggling with math (especially under the anxiety of a big tournament) doesn’t make you a bad person.  People have lost tournaments, including world-level events, due to addition mistakes.  They’re not bad people, and they were embarrassed, but putting in some math practice will help you avoid that same circumstance.

Nobody is born being “good at math”, and nobody is born being “good at archery”.  But we practice archery, get training and advice, and share tips to help each other.  We’re patient with archers who are at different skill levels, and we support each other’s growth and gaining in expertise.

The same thing goes with math.

Also, I’m going to keep using ‘math’ and ‘arithmetic’ interchangeably.  They’re note the same thing, and it’s important why not, but not today.  And one of them is much easier to type and read.

But, My Calculator!

The best tools still requires the operator to know how to use them, and to not make mistakes.  The electronic scoring tablets are a huge help, but people make mistakes with them during every single tournament.  So the manual math is just as important.

Mental Fatigue:

Physical fatigue, stress, and anxiety all cause fatigue in mental ability.  So as you’re doing mental math in a tournament, be aware that you’re far more likely to make mistakes in the later ends.  (For me, it’s always around end 18 of an indoor 600 -TK)  Make more use of double-checking.

Archery Math:

The good news is that there are only a few types of math problems in archery.

  • Adding arrow scores into a per-end score, usually 3 arrows, 5 arrows, or 6 arrows.
  • Adding the per-end score onto the running total.
  • Adding up the price of multiple bbq sandwiches for lunch.

So all of our focus can go onto increasing our confidence with mental addition.  We can focus on adding 3, 5, or 6 numbers that will be between 0 and 10.  And we can more generally practice adding 2-digit numbers and 3-digit numbers.  We’re not trying to win a Fields Medal for Riemann Surfaces like Maryam Mirzakhani.  But she might have trouble earning her red pin…

Foundational Tools:

  • Addition
    • Memorization – For some folks, just memorizing the single-digit math for addition is the right way.  7+5 just “is” 12, and there’s no intermediate step.
    • Counting – For some folks, just quickly counting (even quietly out loud) really is the best way to do accurate single digit math.  There is nothing wrong with that at all.  Try to be nice and not interrupt them when they’re doing that — it will just slow down the game.
  • Subtraction
    • It comes in useful for some of the techniques we’ll talk about below.  It’s also useful for double-checking your answer.  Memorization and/or counting work well here, too.
  • Multiplication
    • Because we’re adding 3 arrows, or 5 arrows, or 6 arrows, it’s helpful to also practice multiplying by 2, 3, 4, 5, & 6.  Memorization is a strong method here, too, but there are a lot of ways to memorize — choose one that works for you.
  • Flashcards
    • Flashcards are the best tool here for most people.  Med students, sales executives, military, astronauts — everyone who needs to remember a pile of information and keep it accurate uses flashcards.  There’s science behind it.
    • We live in a golden age for flashcards.  Anki, Quizlet, and other websites and apps make it easy to create study cards for any topic in the world, and finding pre-made flashcards for addition and multiplication is easy.  Here’s one I found in 5 seconds.  I’m sure there are even better ones out there, too.  Don’t feel weird if the tool you end up liking is aimed at elementary school children.  I use 4th grade computer tools to teach adults to write computer code.  Some of those are great tools, despite the cartoon animals.
    • Spend some time each week doing math flashcards as part of archery practice.
      • Do them in the car on the way to or from the range.  (If you’re driving, have someone read them out loud — or maybe the flashcard apps already do that…)
      • Add it into your team’s warm up routine (“stretch and repeat the 6 times-table!:  6-12-18-24…48, 54, 60”)
    • Study time isn’t graded, it doesn’t have to be fast.  It’s about increasing accuracy and confidence that you’re accurate.

Strategies

Adding 3 numbers

3 numbers between 0 and 10 are a manageable for flash card study.  Break them up into adding the first 2 numbers between 0 and 10 (there are 66 possible combinations).  Then you’ll always have a ‘big’ number between 0 and 20, and the 3rd number to add onto it.  This is mostly the same as adding the first two numbers, but with sometimes needing to carry a 1 into the 10s column.

Put in the hours to become a rockstar at adding 3 numbers like this, and you’re most of the way there for archery math.

Adding 5 or 6 numbers

Because arrows are only ever worth up to 10 points (except in the Lancaster Classic), and because we always score them from highest value to lowest value, the numbers are usually small and in a convenient order.

Try to find a smaller group of numbers that form a pattern that’s easier to manage.  Then switch to counting if necessary.  Or do the first three, then the second three

  • 7+7+6+5+3+1 is a mess
    • 7+7+6, then 5+3+1 is easier.  20 + 9 = 29

Find Patterns that you like

Sometimes certain number patterns are easier than others.  When the numbers are one of those patterns, you’ll already know the answer, or how to easily and confidently solve it.

  • all of the arrows having the same number can be easy.
  • Sometimes you’ll just have favorite numbers that work better.  If you like adding anything and everything to a 3 or to a 7, then great.

Subgroup:

Try to find a smaller group of numbers that form a pattern that’s easier to manage.  Aim to great 10s and 5s.

  • 7-6-6-6-2-1:  The three 6s become a multiplication (3×18), then the 7,2,&1 become 10.  10+18 is easier.

Subtract Down

For scores with a lot of 10s, 9s, and 8s, it’s often easier to start from a perfect score and subtract down the number of points not awarded.  So with arrows of 10, 9, & 9, it’s easier to calculate “That’s 30 points minus 2 points, 28”.  With 9,8,7: that’s 1+2+3 = 6, and 30 minus 6 is 24.

With 6 numbers (and an excellent archer), you might get 10-9-9-9-8-7.  That’s three 1-point-downs, one 2-point-down, and one 3-point-down.  (3×1)+2+1 = 3+2+1 = 6.  And 60-6 is 54.

Depending on the person and the numbers, sometimes subtracting down is easier.

Multiply and Adjust

Sometimes the numbers are almost convenient for multiplication, but not quite.  So pretend they are convenient, then adjust the result.

  • 8-8-7 is almost 8-8-8.  Three 8s is 24, easy multiplication.
    • So add 1 to the 7, do the easy mental multiplication to get 24, then because you added 1, adjust the result by subtracting the 1 back out to get 23.
  • 9-8-8-8-8 is almost 5×8, which would be easy multiplication.
    • So subtract 1 from the 9 to get the easy result of 5×8=40, then adjust and add that 1 back to the result to get 41.

Series of Numbers:

This is similar to multiply-and-adjust.  If you have a series of numbers in a row, the result from adding those numbers is the same as multiplying the middle number by the number of things in the series.

  • 8+7+6:  If we take 1 away from the 8 and add it to the 6, we get 7+7+7.
    • That’s an easier problem to solve, and it’s clear that it’s 21
  • 5+4+3+2+1:  This works the same way with a longer series of numbers in a row.
    • There are five numbers, and the middle number is 3.  So this is 5×3=15
  • 7+5+3: This also works for numbers that aren’t exactly next to each other, but they are the same amount further apart.  It’s still 3×5=15
  • 10+9+8+7: This one is trickier because there’s an even number, but it uses the same idea.
    • When there’s an even number, you use the average of the two middle numbers
    • 4×8.5=34  (Feel free to multiply-and-adjust this as “leave off the 0.5, 4×8=32, now add back on 4×0.5=2” –> 32+2=34.)
    • It’s rarer that that’s useful, but it could help with a score of 10-9-8-7-5-5 to group the first 4 digits (now we know they’re 34) then adding two 5s is easy because they’re a 10 to get 44.
  • 10+8+6+4+2+0: It still works with the skipping-numbers idea, too, as long as they’re the same distance apart:
    • Take the average of the middle numbers (6 and 4 average to 5), then multiply by the number of digits (6)
    • 5×6=30 – Which you can also see by adding the 10 + (8+2) + (6+4)

Double-Checking

It’s important to double-check your answers, especially when you’re not confident in your answer (or someone else is not confident in your answer).  The best way to double-check is to switch to a different math approach (add up instead of subtracting down, or multiply-and-adjust).  If you get the same answer from two different methods, it’s probably correct.  If not, then swallow your pride and just ask others in the group to verify your math.  This is a good step to take when you’re feeling mental fatigue.  Every archer would rather double-check the math than get the score wrong on their scorecard.

Other Strategies

If you have other strategies that you find helpful, I’d love to hear them, and I’ll post them here so other folks can benefit from them, as well.

Special Note: Dyscalculia – “Dyslexia for numbers”

There are people who have a true learning difficulty for numbers and arithmetic.  This is a real developmental disorder with physical, observable differences in brain structure.  People living with this condition often take longer working with numbers and may be more prone to making mistakes in calculations.  This isn’t “being bad at math”, and people living with dyscalculia deserve sympathy and  patience.  For archery purposes, people with diagnosed dyscalculia may be better volunteering to call arrows on a target instead of writing scores.  Regardless of that, everyone can benefit from math practice.