Algorithms and Number Patterns: a week of ready-to-teach maths

Five days of lessons for Year 6 Algebra. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet, plus step cards and target cards, machine cards and input cards, a mini-check and every answer.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Cut out the step and target cards (Days 1, 2 and 5) and the machine and input cards (Day 3) once, one set per pair.
4. Open the free interactive unit on your board. Every plan tells you which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and answers are all on the one page.

No maths background needed

This pack is written for the busy generalist teacher. Each plan explains the idea in plain words, lists the misconceptions children bring, and gives the answers. The big idea of the week: an algorithm is a list of steps carried out in order, and the same steps always give the same result. A loop repeats steps until a stop rule says finish; a decision sends a number one way or another. Running an algorithm many times makes patterns appear, and explaining why a pattern appears is real mathematics.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 50 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point if your timetable runs two shorter sessions.

Words to watch

A loop sends you back to an earlier step; a stop rule ends the loop. A decision asks a yes or no question, such as “Is the number even?”, and each answer leads to a different step. The 3n + 1 algorithm on Day 4 is a famous puzzle: nobody has proved it always reaches 1, so children are doing real exploring.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 follows algorithms written as steps and as a flowchart; Day 2 uses them to predict; Day 3 adds decisions; Day 4 runs algorithms many times so patterns emerge; and Day 5 designs new algorithms and explains a pattern. It builds on the Year 5 algorithms that test factors, multiples and divisibility (AC9M5N10) and the AC9M6A01 unit on growing patterns. It leads to the Year 7 algorithms that sort and classify shapes (AC9M7SP04) and to the loops and branches of Digital Technologies programs.

Materials for the week (one trip)

• From the classroom: scissors, pencils, this pack printed, and a calculator for each pair on Day 4.
• Cut out once: the step and target cards, and the machine and input cards, one set of each per pair.
• If you have devices, a block-coding tool is a good extra on Day 5 for turning an algorithm into a program.

Words for the word wall

algorithm, step, input, output, rule, loop, repeat, stop, decision, flowchart, sequence, set, pattern, predict, explain

Dear families

This week in maths, Year 6 follows and designs algorithms: lists of steps, some with loops and decisions, that make sets of numbers. We run them many times, look for the patterns that appear, and try to explain them.

Try this at home

• Write the steps for a daily routine as an algorithm. Where is there a decision, and where is a loop?
• Pick a number. If it is even, halve it; if it is odd, multiply by 3 and add 1. Keep going. Do you reach 1?
• Add the digits of 18, 27 and 36, and of other multiples of 9. What do you notice?
• Add the odd numbers 1, 3, 5, 7, ... and say the totals. What kind of numbers are they?

My algorithms this week

Fill one row a day. Write an algorithm you used, the numbers it made and a pattern you saw.

Printed from the free seegongsik Algorithms and Number Patterns teaching pack · seegongsik.com/au/y6/algebra/AC9M6A03/pack

Follow the steps

An algorithm is a list of steps carried out in order; following the same steps always gives the same numbers. Children follow written algorithms and a flowchart, notice the loop that repeats a step and the stop rule that ends it, and write an algorithm of their own for a given set.

We are learning to

• follow an algorithm step by step to make a set of numbers,
• read an algorithm written as a flowchart,
• write the steps of an algorithm that makes a given set.

Success criteria

• I can follow ‘start at 100, take away 12’ to its stop rule and list the numbers.
• I can write an algorithm that makes 4, 11, 18, 25, 32.

You need

The worksheet, one per child, and the step and target cards (cut-outs), one set per pair.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the flowchart.

Watch for these ideas

• Doing the operation before writing the start number down, so the set loses its first number.
• Checking the stop rule at the wrong moment, so one number too many or too few is written.
• Missing the loop: doing step 3 once and stopping.

Answers

• Part A: 2, 5, 8, 11, 14, 17; 100, 88, 76, 64, 52, 40; 1, 2, 4, 8, 16, 32, 64.
• Part B: 5, 9, 13, 17, 21, 25, 29.
• Part C, one algorithm that works: Start at 4. Write the number down. Add 7. If you have written 5 numbers, stop. Otherwise go back to step 2.

Follow the steps

A. Follow each algorithm

Do the steps in order. Write every number you write down in step 2.

B. Follow the flowchart

C. Write the algorithm

Write the steps of an algorithm that makes 4, 11, 18, 25, 32.

1. ______________________________   2. ______________________________

3. ______________________________   4. ______________________________

Rules and predictions

Once you know how an algorithm builds its numbers, you can predict without running it. Children run two rules from the same start, find far-off numbers by counting the steps, decide whether a number will appear, and see the shape a set makes on a number grid.

We are learning to

• compare the sets two rules make from the same start,
• predict a far-off number without listing every number,
• decide whether a number belongs to the set an algorithm makes.

Success criteria

• I can find the 100th number of ‘start at 6, add 5’ without listing them.
• I can explain whether 2026 is in that set.

You need

The worksheet, one per child, coloured pencils, and the step and target cards (cut-outs), one set per pair.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the predictions in Part B.

Watch for these ideas

• Counting the jumps wrongly: the 10th number has 9 jumps after the start.
• Thinking the doubling rule stays behind because it starts slowly.
• Deciding a number is in the set because it lies between two of its numbers.

Answers

• Part A: add 4: 3, 7, 11, 15, 19, 23; double: 3, 6, 12, 24, 48, 96. The doubling rule passes 50 at its 6th number (96).
• Part B: 51 and 501. 2026 is in the set, the 405th number: 2026 − 6 = 2020 is a multiple of 5 (404 jumps). 100 is not in the start 7, add 4 set, as 100 − 7 = 93 is not a multiple of 4; the last number below it is 99.
• Part C: 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49. Each is one more than a multiple of 3, so they fall in slanted lines on a grid ten wide.

Rules and predictions

A. Two rules, one start

Rule 1: start at 3 and add 4. Rule 2: start at 3 and double. Write six numbers for each.

Which rule passes 50 first, and at which number? ______________________________

B. Predict without listing

1. Start at 6 and add 5. What is the 10th number? ______ The 100th? ______

2. Is 2026 in that set? How do you know? ________________________________________

3. Start at 7 and add 4. Is 100 in the set? ______ What is the last number below 100? ______

C. A set on a grid

Start at 1, add 3, and stop when the number is more than 50. Colour every number you write.

What pattern do the coloured squares make? Why? __________________________________________

Decisions

A decision step asks a yes-or-no question and sends the number down one of two paths. Children run machines with a decision, run one machine again and again to make a chain, and design a machine of their own that turns given inputs into given outputs.

We are learning to

• run an algorithm with a decision step,
• run an algorithm again and again on its own output,
• design an algorithm with a decision to match given inputs and outputs.

Success criteria

• I can run machine 1 from 13 all the way to 1.
• I can design a machine that turns 10 into 5 and 7 into 21.

You need

The worksheet, one per child, and the machine and input cards (cut-outs), one set per pair.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the chains in Part B.

Watch for these ideas

• Following the wrong path, or doing both steps, after the decision.
• Treating ‘more than 20’ as if it included 20.
• Forgetting to feed each output back in as the next input.

Answers

• Part A: machine 1: 12 → 6, 7 → 8, 30 → 15, 15 → 16; machine 2: 25 → 5, 8 → 17, 40 → 20, 20 → 29.
• Part B: 13 → 14 → 7 → 8 → 4 → 2 → 1 (6 steps); 20 → 10 → 5 → 6 → 3 → 4 → 2 → 1 (7 steps).
• Part C, one machine that works: If the number is even, halve it. Otherwise, multiply it by 3.

Decisions

A. Run the machines

B. Again and again

Put machine 1’s output back in as the next input. Keep going until you reach 1. Count the steps.

Start at 13: ______________________________________________   steps: ______

Start at 20: ______________________________________________   steps: ______

C. Design a machine

Write a machine with a decision that turns 10 into 5 and 7 into 21.

If the number is ____________________, ____________________. Otherwise, ____________________.

Test it on another input of your own: ______ → ______

Patterns that emerge

Running an algorithm many times can make a pattern appear that nobody planned. Children run the 3n + 1 algorithm from many starts and the digit-sum algorithm on many numbers, record what happens in a table, and explain the patterns they see.

We are learning to

• run an algorithm many times and record the results in a table,
• spot patterns that emerge from the results,
• explain why a pattern happens, not just that it happens.

Success criteria

• I can explain why starting at 12 takes one more step than starting at 6.
• I can predict the digit sum of a multiple of 9 and say why.

You need

The worksheet, one per child, and a calculator for each pair for the longer chains.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the class table of chains.

Watch for these ideas

• Halving an odd number, or multiplying an even one by 3, at the decision.
• Counting the start number as a step.
• Calling a pattern true after one or two examples, without a reason.

Answers

• Part A, the steps to reach 1: 2 takes 1, 3 takes 7, 4 takes 2, 5 takes 5, 6 takes 8, 8 takes 3, 10 takes 6, 12 takes 9, 16 takes 4.
• Part B: 2, 4, 8 and 16 only ever halve: they are the numbers you get by doubling from 1, so halving walks straight back down. Doubling a start adds exactly one step (2 and 4, 3 and 6, 4 and 8, 5 and 10, 6 and 12, 8 and 16), because the first step halves it back to the smaller start. Past 1 a chain would loop 4, 2, 1 for ever.
• Part C: 18 → 9, 27 → 9, 36 → 9, 99 → 18 → 9, 189 → 18 → 9; 14 → 5, 23 → 5, 32 → 5, 41 → 5. Multiples of 9 end at 9; adding 9 adds 1 ten and takes 1 one, so the digit sum stays the same.

Patterns that emerge

A. Run it from many starts

B. Look for patterns

1. Which starts reach 1 by halving every time? What do they have in common? ______________________________

2. Compare the steps for 3, 6 and 12, and for 5 and 10. What happens when you double the start? Why?

______________________________________________________________________________

______________________________________________________________________________

C. Digit sums

Add the digits, then add the digits of the answer, until one digit is left.

What do you notice? Can you explain it? ______________________________________________

Design and explain

Children explain a pattern that emerges from an algorithm, then design algorithms of their own, one with a loop and one with a decision, to make given sets. A partner tests each design by following it exactly, the way a computer would.

We are learning to

• explain an emerging pattern with a picture or a reason,
• design an algorithm with a loop, and one with a decision, to make a given set,
• test an algorithm by following it exactly.

Success criteria

• I can show with dots why adding odd numbers makes square numbers.
• I can write an algorithm that makes the odd multiples of 3 up to 45.

You need

The worksheet, one per child, and the step cards (cut-outs), one set per pair.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the designs in Part B.

Watch for these ideas

• Writing a step a reader could do two ways, such as ‘keep adding’ with no number.
• Leaving out the stop rule, so the loop never ends.
• Writing a number down before the decision has tested it.

Answers

• Part A: 1, 4, 9, 16, 25, 36, the square numbers; each odd number adds an L of dots round the last square, making the next square.
• Part B (other designs that make the set are fine): (1) Start at 50. Write the number down. Take away 5. If the number is less than 5, stop. Otherwise go back to step 2. (2) Start at 3. If the number is odd, write it down. Add 3. If the number is more than 45, stop. Otherwise go back to step 2.
• Part C: a design passes if a partner, following it exactly, writes exactly the set and then stops.

Design and explain

A. Add the odd numbers

1. Set the total to 0 and the odd number to 1. 2. Add the odd number to the total and write the total down. 3. Add 2 to the odd number. 4. If you have written 6 totals, stop. Otherwise go back to step 2.

Totals: ____________________________________________   What kind of numbers are they? ____________

Use the dot pictures to explain why the totals are always this kind of number. ________________________

______________________________________________________________________________

B. Design two algorithms

1. With a loop, make 50, 45, 40, 35, 30, 25, 20, 15, 10, 5.

1. ______________________________   2. ______________________________

3. ______________________________   4. ______________________________

2. With a decision, make the odd multiples of 3 up to 45: 3, 9, 15, 21, 27, 33, 39, 45.

1. ______________________________   2. ______________________________

3. ______________________________   4. ______________________________

C. Test a partner’s design

Follow your partner’s algorithm exactly. Numbers it made: ____________________________________

Does it make the set? If not, which step needs fixing? __________________________________________

Step cards and target cards

Cut out the step cards and the target cards once, one set for each pair. Write numbers in the blanks in pencil.

Algorithm builder (Days 1, 2 and 5)

1. Turn over a target card. Lay out step cards in order to make an algorithm for that set, and fill the blanks.
2. Your partner follows your algorithm exactly and writes down the numbers it makes.
3. If the numbers are the target set and the algorithm stops, keep the target card. Swap roles.

Teacher note, one algorithm for each target: (1) Start at 5. Write the number down. Add 4. If you have written 5 numbers, stop. Otherwise go back to step 2. (2) Start at 60. Write the number down. Take away 7. If the number is less than 20, stop. Otherwise go back to step 2. (3) Start at 3. Write the number down. Double it. If the number is more than 100, stop. Otherwise go back to step 2. (4) Start at 256. Write the number down. Halve it. If you have written 6 numbers, stop. Otherwise go back to step 2. (5) Start at 1. Write the number down. Add 2. If the number is more than 15, stop. Otherwise go back to step 2. (6) Start at 0. Write the number down. Add 25. If you have written 5 numbers, stop. Otherwise go back to step 2. (7) Start at 2.5. Write the number down. Add 2.5. If you have written 4 numbers, stop. Otherwise go back to step 2. (8) Start at 99. Write the number down. Take away 11. If you have written 5 numbers, stop. Otherwise go back to step 2.

Machine cards and input cards

Cut out the six machine cards and the twelve input cards once, one set for each pair.

Machine race (Day 3)

1. Lay the machine cards face up. Shuffle the input cards face down.
2. Turn over an input card. Take turns to choose first: each player picks a different machine and runs the input through it.
3. The bigger output wins the input card. If the outputs are equal, the card goes to the bottom of the pile.

Teacher note: every output.

What we know: Algorithms and Number Patterns

Work on your own. Show your thinking if you can.

1. Follow: Start at 4. Write the number down. Add 6. If you have written 5 numbers, stop. Otherwise go back to step 2. ______________________________
2. Follow: Start at 81. Write the number down. Take away 9. If the number is less than 40, stop. Otherwise go back to step 2. ______________________________
3. What comes next: 7, 12, 17, 22, ______
4. If the number is even, halve it. If it is odd, add 5. What comes out for 18 and for 9? ______ ______
5. Start at 6 and add 7. Will 100 be in the set? How do you know? ______________________________
6. Run the 3n + 1 algorithm from 20. How many steps does it take to reach 1? ______
7. Write an algorithm that makes 3, 8, 13, 18, 23. ______________________________
8. The totals of the odd numbers go 1, 1 + 3, 1 + 3 + 5, ... Write the next two totals. What kind of numbers are they? ______________________________

Answers and marking guide

Answers

1. 4, 10, 16, 22, 28.
2. 81, 72, 63, 54, 45.
3. 27.
4. 9 and 14.
5. No: 100 − 6 = 94 is not a multiple of 7; 97 and 104 are in the set.
6. 7 steps: 20, 10, 5, 16, 8, 4, 2, 1.
7. Start at 3. Write the number down. Add 5. If you have written 5 numbers, stop. Otherwise go back to step 2. (other algorithms that make the set are fine)
8. 16 and 25; the square numbers.

A quick three-level guide

Eight questions, four ideas. A child at standard answers most questions and can say what each step of an algorithm does.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: following steps, rules and predictions, decisions, emerging patterns, and designing and explaining.