Choose which steps go up
A route to (3, 2) is 5 steps: 3 rights, 2 ups
Choose which 2 of the 5 steps are ups
5 × 4 ways in order, ÷ 2 because order doesn't matter
C(5, 2) = 10
Counting routes on a grid How many shortest routes cross a grid, and why is the answer a choose number?
Maths olympiad · 45 minutes
Students count routes across a grid by hand, discover that each corner adds the two before it, find Pascal's triangle, and only then name the shortcut: choose. Slides, a vote that shows who's stuck, problems that climb to olympiad level, and short written arguments to mark.
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Move n. The marked corners are n blocks from home at (0, 0), walking only right or up along the grid. Start with n = 1, 2 and 3: how many shortest routes reach each marked corner, and what do they add up to? Predict the total for n = 6, and say why it must be so.
Whole-number points: (0, 3), (1, 2), (2, 1), (3, 0)
The idea in one minute
A shortest route on a grid only goes right or up. Every route into a corner arrives from the corner on its left or the corner below it, never both, so the number of routes to a corner is the sum of those two numbers. Written on the grid, the numbers make Pascal's triangle. A route to (a, b) is a + b steps, b of them up, so the count is the number of ways to choose which b steps go up: C(a + b, b).
The lesson goes problem first. Students count routes to one corner by hand and argue about whether they found them all. They then fill the grid by adding, see Pascal's triangle appear, and only then is the shortcut named and explained. The same two ideas (add when routes split into separate kinds; multiply for one part and then another) answer routes through a point and routes that avoid one. Practice climbs from a 2 by 2 grid to problems at olympiad level, with short written explanations the teacher marks.
By the end, students can
- count routes by adding at each corner, and explain why adding counts nothing twice
- see Pascal's triangle in the grid and use C(a + b, b) as the shortcut
- count routes through a point by multiplying, and routes that avoid it by subtracting
- write a short, clear argument for a count
Key words
- lattice path
- shortest route
- Pascal's triangle
- choose, C(n, k)
- multiplication principle
- complement
Before this lesson
Students should multiply and divide whole numbers confidently and read coordinates. No counting rules are assumed: the lesson builds them.
Next: Pick's theorem, Whole-number solutions of ax + by = c.
The lesson, step by step
60 minutes in total. Each step says what changes for this length.
5 min 1 · Hook · students think
Home to school
Students count shortest routes to one corner by hand before anyone explains.
Say“Home is at (0, 0) and school at (3, 2). You walk along the grid lines, only right or up. How many shortest routes are there? Write them all down, then convince your neighbour you have them all.”
Ask- How do you write a route down quickly?
- How do you know you haven't missed one, or written one twice?
Watch forStudents who draw routes at random and stop at 6 or 8: ask for a system, such as all routes that start with up first.
For the teacher
Give three minutes. Most draw routes; a few invent a code such as RRURU (right, right, up, right, up).
Collect the totals on the board. Ask how anyone could be sure. Don't settle it yet: the next step gives a method that settles it.
Take the class's answers without saying which is right. A spread of answers (8, 10, 12) is the point: we need a way to count that can't miss or repeat.
Shorter: Ask for the count with a code (R and U) at once; skip the drawing round (−2 min).
12 min 2 · Explain · you explain
Discover it, then name it
Fill the grid by adding, find Pascal's triangle, then explain the choose shortcut.
Say“Every route into a corner arrives from the left or from below. So the routes to a corner are the routes to its left neighbour plus the routes to the one below.”
Ask- Why is every corner on the bottom edge a 1?
- Could a route arrive from the left and from below at once? So is anything counted twice?
- Turn the grid 45 degrees. What do the numbers make?
Watch for5 × 4 = 20 for the choose step: picking step 2 then step 4 is the same as picking 4 then 2, so divide by 2.
Through a point: multiply
(0, 0) to (3, 2) through (1, 1)
2 ways to (1, 1); from there 3 ways on
Each way in goes with each way on: 2 × 3 = 6
Avoid a point: subtract
(0, 0) to (3, 2), not through (1, 1)
All routes take away the ones through it
10 − 6 = 4
For the teacher
First slide: fill the grid. The corners along the bottom and up the left side are 1: one way only. Every other corner is the corner to its left plus the corner below, because a route arrives from exactly one of them. The hook's answer appears: 10 at (3, 2).
Second slide: turn the grid so that (0, 0) is at the top. The numbers are Pascal's triangle. Students say what the rows add up to (1, 2, 4, 8, ...), and why: the routes of n steps, ending anywhere, are 2 × 2 × ... × 2.
Third slide: name it. A route is a word of R's and U's: to (3, 2), 3 R's and 2 U's. Choosing where the 2 U's go fixes the route, so there are C(5, 2) routes: 5 × 4 in order, divided by 2 because the order of picking doesn't matter. In general, routes to (a, b) number C(a + b, b).
Fourth slide: through a point, multiply (each way in with each way on); to avoid a point, take the routes through it from all of them.
Ask the class to fill the grid before you show the numbers. When someone says the grid is Pascal's triangle, ask why it should be: each number in the triangle is the two above it added, as each corner is the two before it.
Shorter: Fill the grid and name C(a + b, b); show through and avoid as one slide to read (−4 min).
8 min 3 · Your turn · students write
Worked, then yours
Each worked example beside a twin that students solve themselves.
Say“Watch me do one. Then you do its twin. Count every one two ways: by adding on a grid, and by choosing.”
Ask- Before you calculate: will it be more or fewer than the last one? Why?
Watch forAdding the ways in and the ways on when the route goes through a point: each way in goes with each way on, so multiply.
Choose
Routes to (4, 2): 6 steps, 2 of them up
C(6, 2) = 6 × 5 ÷ 2 = 15
Then theirs
Routes to (3, 3)
Through a point
(0, 0) to (4, 3) through (2, 1)
To (2, 1): C(3, 1) = 3; then 2 right, 2 up: C(4, 2) = 6
3 × 6 = 18
Then theirs
(0, 0) to (3, 3) through (1, 2)
Avoid a point
(0, 0) to (4, 3), not through (2, 1)
C(7, 3) = 35 in all
35 − 18 = 17
Then theirs
(0, 0) to (3, 3), not through (1, 2)
For the teacher
Work the first example aloud, writing each line. Then students solve its twin, by a grid or by choosing, and check it the other way.
Do the same for each pair. The third pair uses the second: the routes through the point are the ones to take away.
Open the answers slide once most have finished their twins. Ask someone who filled a grid and someone who chose to compare: the same number two ways is the check.
Shorter: Do the first two pairs only (−3 min).
25 min 4 · Who's stuck? · students vote
Vote, then climb
One question shows who's stuck; then problems that climb from fluency to olympiad level.
For this length: Question 9 written up in full, then read aloud and improved together
Say“Choose an answer on your own. Nobody talks until everyone has chosen. Then work up the problems as far as you can; the last ones are olympiad level.”
Ask- Can you check your answer by filling a small grid?
Watch forUp steps chosen in order and not divided; every string of steps counted, wherever it ends; the coordinates multiplied.
Routes go along grid lines, only right or up, from (0, 0). Show your working on paper. Questions 6 and 9 need a written explanation.
The question that shows who's stuck
- How many shortest routes are there from (0, 0) to (5, 2)?
In your Classroom, as they answer (an example) - A11
- B8
- C4
- D3
Each wrong answer is a different mistake. While they answer, your Classroom shows who chose which, so you know in seconds whom to help with what.
- How many shortest routes are there from (0, 0) to (2, 2)?
- Row 6 of Pascal's triangle: 1, 6, ___, ___, 15, 6, 1
- How many shortest routes are there from (0, 0) to (5, 3)?
- How many different 7-letter words can be made from four A's and three B's (such as AABABBA)?
- How many shortest routes from (0, 0) to (5, 3) pass through (2, 2)?
- Explain why the number of routes to a corner is the number of routes to the corner on its left plus the number to the corner below it. Why is no route counted twice, and none missed?
- How many shortest routes from (0, 0) to (4, 4) avoid the corner (2, 2)?
- How many shortest routes from (0, 0) to (4, 4) never go above the line y = x? (They may touch it.)
- Stretch: every route from (0, 0) to (3, 3) crosses the line x + y = 3 at exactly one corner. Use this to explain why 1² + 3² + 3² + 1² = 20. Then say what 1² + 4² + 6² + 4² + 1² must be, and why.
Support · For question 1: draw the 2 by 2 grid, put 1 on every corner of the bottom edge and the left side, then add. For the olympiad questions: make the grid smaller, count by adding, and look for the pattern before you prove it.
For the teacher
Everyone chooses before anyone explains. Then ask a student for each answer to say how they got it, and to try the same method on routes to (2, 2), which the class can list in full. Choosing in order, strings that end anywhere and multiplied coordinates all show up here.
Questions 1–3 build fluency; 4–5 give the same idea a new look; 6 asks for a written reason; 7–8 are competition style; 9 is olympiad level and needs a short proof.
Students show their working on paper or the board; the quiz keeps their final answers, and the written ones come to you to mark.
If the class is split, list the routes to (2, 2) together as words of R's and U's (there are few enough), then ask each student to try their method on that small case. Long answers (6, 9) are marked by you: one mark for the idea, one for why nothing is counted twice or missed, one for the conclusion. Model answers are in each question's explanation.
Shorter: After the vote, questions 1–5 only (−6 min).
Longer: Question 9 written up in full, then read aloud and improved together (+10 min).
10 min 5 · Exit · students show
Count it, and say why
One count with a written reason shows who can count without missing or repeating.
For this length: Swap answers and mark a partner's reason: is anything missing?
Say“Count the routes, any way you like. Then explain in two sentences why you have counted every route exactly once.”
Watch forA right number with no reason: the reason is what an olympiad marks.
A number and two sentences.
- How many shortest routes from (0, 0) to (4, 3) avoid the corner (1, 1)? Explain in two sentences why you have counted every route exactly once.
For the teacher
Routes from (0, 0) to (4, 3) that avoid the corner (1, 1).
A full answer says what is counted, how, and why nothing is counted twice.
Read the reasons, not only the numbers: start the next lesson with the best one (anonymous) and one that needs improving. Set the homework.
Longer: Swap answers and mark a partner's reason: is anything missing? (+5 min).
One to one: let the student fill the grid themselves, corner by corner, saying aloud why each number is the sum of two others; then ask them to predict a count before checking it with the choose shortcut.
Homework: count and explain
Routes go along grid lines, only right or up. Count each one two ways where you can (adding on a grid, and choosing). Question 6 is a written explanation: write it as you would in an olympiad.
- How many shortest routes are there from (0, 0) to (3, 4)?
- Write out row 7 of Pascal's triangle. What do its numbers add up to, and why?
- How many shortest routes from (0, 0) to (6, 2) pass through (3, 1)?
- How many shortest routes from (0, 0) to (5, 5) avoid the corner (2, 2)?
- In 3D, a route moves one unit at a time in the x, y or z direction, never backwards. How many shortest routes go from (0, 0, 0) to (2, 1, 1)?
- Explain why there are as many routes to (a, b) as to (b, a), without working either number out.
Print the worksheet with working space and homework. Teacher answers come with your Library copy.
Teaching counting routes on a grid: common questions
Why does Pascal's triangle appear when you count routes on a grid?
Every route into a corner arrives from the corner on its left or the corner below, never both, so the corner's count is the sum of those two. Pascal's triangle is built by the same rule: each number is the two above it added. Turn the grid so that the start is at the top and the two pictures are the same.
Should I teach the choose formula first or the adding method?
The adding method first. It explains why the counts work, it handles blocked corners and odd shapes the formula can't, and students discover Pascal's triangle themselves. The formula C(a + b, b) then arrives as a shortcut they understand, with the divide-by-order step explained by double counting.
Which competitions use counting problems like these?
Routes on grids, arrangements of letters and counting with restrictions come up in junior and intermediate maths challenges and their olympiad rounds, and in olympiads at every level. The problems in this lesson are written for it; it uses no past papers.
How do I mark the written questions?
Look for the idea, not the length: what is counted, how it is split into cases or steps, and why nothing is counted twice or missed. A teacher's copy has a model answer for each written question with what full marks needs. Read the best answers aloud next lesson: students learn to write by seeing good writing.
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