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

Five days of lessons for Year 5 Number. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet, plus flowchart cards to build an algorithm, four hundred squares for shading, 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 flowchart cards once, one set per pair. Print the hundred squares, one sheet per child.
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 precise list of steps, often with yes-or-no decisions, that anyone can follow to the same result. Running algorithms on many numbers makes patterns appear: a number whose digits add to a multiple of 9 is divisible by 9, the multiples of 9 fall on a diagonal of the hundred square, and only square numbers have an odd number of factors.

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.

Calculators

On Days 1, 2 and 5 a calculator checks the results: a whole-number answer to N ÷ 3 means N is divisible by 3. The algorithms themselves are run by hand, so children see every step and decision. One calculator per pair is enough.

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

Days 1 and 2 follow algorithms that test for divisibility; Day 3 builds algorithms of the children’s own, including one that finds every factor; and Days 4 and 5 run algorithms on many numbers to make patterns appear and test them. It builds on Year 4’s algorithms that generate sets of numbers (AC9M4N09) and this year’s factors, multiples and divisibility (AC9M5N02), and it leads to Year 6, where children create algorithms that use rules to generate sets of numbers and explain the patterns (AC9M6A03).

Materials for the week (one trip)

• From the classroom: scissors, glue, coloured pencils (two colours each), mini whiteboards, this pack printed.
• For Days 1, 2 and 5: one calculator per pair.
• Cut out once: the flowchart cards (Day 3). The hundred squares (Day 4) stay whole, one sheet per child. No maths equipment to buy.

Words for the word wall

algorithm, step, decision, flowchart, input, output, divisible, digit sum, factor, multiple, common multiple, pattern

Dear families

This week in maths, Year 5 follows and builds algorithms: lists of steps, with yes-or-no decisions, that always give the same result. One tests whether a number is divisible by 3: add its digits, and if the sum is a multiple of 3, so is the number. 582 passes, because 5 + 8 + 2 = 15.

Try this at home

• Test the year you were born, or a house number, for 3 and 9 with the digit sum. Check on a calculator.
• Write the steps of an everyday task, such as making toast, as an algorithm with one decision.
• Start at 0 and keep adding 7. Look at the last digits. When do they repeat?
• Ask your child why 36 has an odd number of factors.

My algorithms this week

Fill one row a day. Test a number from home, or write down a pattern you found.

Printed from the free seegongsik Number Patterns and Algorithms teaching pack · seegongsik.com/au/y5/number/AC9M5N10/pack

Follow the flowchart

Children follow an algorithm drawn as a flowchart: a start, a step, a yes-or-no decision and two ways out. The digit-sum test for 3 is the first one: add the digits, decide whether the sum is a multiple of 3, and follow the yes or the no. A calculator checks the result.

We are learning to

• follow an algorithm one step at a time, including a yes-or-no decision,
• use the digit sum to test for 3 and for 9,
• check a result by dividing on a calculator.

Success criteria

• I can put 471 through the divisible-by-3 flowchart and say which path it takes.
• I can explain what the decision box does.

You need

One calculator per pair. Mini whiteboards. The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B by running 84 through the flowchart again.

Watch for these ideas

• Testing the number instead of the digit sum: ‘471 is odd, so no’. The decision is about the digit sum.
• Following both ways out of the decision. A number takes the yes or the no, never both.
• Thinking the 3 test works for 9 too. The 9 test needs the digit sum to be a multiple of 9.

Answers

• Part A: 471: 4 + 7 + 1 = 12, yes, 471 ÷ 3 = 157; 1236: 1 + 2 + 3 + 6 = 12, yes, 1236 ÷ 3 = 412; 538: 5 + 3 + 8 = 16, no; 805: 8 + 0 + 5 = 13, no; 999: 9 + 9 + 9 = 27, yes, 999 ÷ 3 = 333.
• Part B: 576: 5 + 7 + 6 = 18, yes, 576 ÷ 9 = 64; 1001: 1 + 0 + 0 + 1 = 2, no; 4302: 4 + 3 + 0 + 2 = 9, yes, 4302 ÷ 9 = 478; 738: 7 + 3 + 8 = 18, yes, 738 ÷ 9 = 82; 512: 5 + 1 + 2 = 8, no. Part C: every multiple of 9 is a multiple of 3 (9 = 3 × 3), but not the other way: 471 passes for 3 (digit sum 12) and fails for 9.

Follow the flowchart

A. Put each number through the flowchart

B. The same flowchart with 9 in the decision

C. What do you notice?

Put your Part B ‘yes’ numbers through the 3 flowchart too. Do they all pass? Does every number that passes for 3 pass for 9? Give an example.

_____________________________________________________________________

Quick tests

Children learn quick tests that decide divisibility without dividing: the last digit for 2, 5 and 10, the last two digits for 4, the digit sum for 3 and 9, and two tests together for 6. Each test is a tiny algorithm with one decision.

We are learning to

• use quick tests for 2, 3, 4, 5, 9 and 10,
• combine two tests: divisible by 6 means divisible by 2 and by 3,
• check a test with a calculator.

Success criteria

• I can test 2340 for 2, 3, 4, 5, 6, 9 and 10.
• I can explain why ‘divisible by 6’ needs two tests.

You need

One calculator per pair. Mini whiteboards. The worksheet, one per child.

Lesson flow (about 50 minutes)

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

Watch for these ideas

• Testing 4 with the last digit only: 714 ends in 4, but 14 is not a multiple of 4. Use the last two digits.
• Thinking an odd number can be divisible by 6. Every multiple of 6 is even.
• Using the digit sum for 2 or 5. The digit sum only tests 3 and 9.

Answers

• Part A: 2340 passes for 2, 3, 4, 5, 6, 9, 10; 735 passes for 3, 5; 1008 passes for 2, 3, 4, 6, 9; 4515 passes for 3, 5; 612 passes for 2, 3, 4, 6, 9.
• Part B: whole-number answers match the ticks. Part C: 34□ by 3: 2, 5, 8; 5□6 by 9: 7; 7□0 by 4: 0, 2, 4, 6, 8; 81□ by 6: 0, 6.

Quick tests

A. Tick the numbers each test passes

B. Check with a calculator

Choose three ticks and two crosses from your table. Divide on a calculator. A whole-number answer means divisible.

C. Missing digits

Write every digit that can go in the box.

34□ is divisible by 3:   ______________________

5□6 is divisible by 9:   ______________________

7□0 is divisible by 4:   ______________________

81□ is divisible by 6:   ______________________

Build an algorithm

Children build algorithms of their own: from flowchart cards, one that decides whether a number is divisible by 6, then a factor finder that tries 1, 2, 3 and so on and stops once the number tried, times itself, is bigger than N.

We are learning to

• put steps and decisions in order to make an algorithm,
• run an algorithm that finds every factor of a number,
• explain when an algorithm can stop.

Success criteria

• I can arrange the flowchart cards into an algorithm that decides whether a number is divisible by 6.
• I can find every factor of 36 with the factor finder and say why I could stop.

You need

The flowchart cards (cut-out sheet 1), one set per pair. Glue. The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after the flowchart; start Session B with the factor finder on 24.

Watch for these ideas

• A decision card with only one way out. Every decision needs a yes and a no.
• Stopping too soon, or never stopping. Stop once the number tried, times itself, is bigger than N.
• Listing a factor twice: 6 × 6 gives the factor 6 once.

Answers

• Part A, one working order: start; last digit even? (no → not divisible); add the digits; digit sum a multiple of 3? (yes → divisible by 6, no → not). Either test can come first. 714 yes (even, digit sum 12); 935 no (odd); 1004 no (even, digit sum 5); 552 yes (even, digit sum 12).
• Part B: 36: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6; stop at 7; 9 factors. 40: 1 × 40, 2 × 20, 4 × 10, 5 × 8; stop at 7; 8 factors. 29: 1 × 29; stop at 6; 2 factors. Part C: 36 has an odd number of factors because 6 pairs with itself; 29 has only two factors, 1 and itself.

Build an algorithm

A. Build a divisible-by-6 flowchart. Glue it here, then run four numbers.

B. The factor finder

1. Start with d = 1.
2. Does d divide N exactly? If yes, write the pair d and N ÷ d.
3. Add 1 to d.
4. Is d × d bigger than N? If yes, stop. If no, go back to step 2.

C. What do you notice?

Why does 36 have an odd number of factors? ____________________________ Why does 29 have only two? ____________________________

Multiples make patterns

Children run ‘start at k, add k’ on a hundred square and shade as they go. Patterns appear: the multiples of 9 make a diagonal and the multiples of 5 make two columns. Where two patterns overlap are the common multiples.

We are learning to

• run ‘start at k, add k’ to shade the multiples of a number,
• describe the pattern the shading makes, and explain it,
• find the common multiples where two patterns meet.

Success criteria

• I can shade the multiples of 9 and describe the diagonal they make.
• I can list the common multiples of 4 and 6 to 100 and say what they have in common.

You need

The hundred squares (cut-out sheet 2), one sheet per child. Coloured pencils, two colours. The worksheet, one per child.

Lesson flow (about 50 minutes)

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

Watch for these ideas

• Slips when adding on. Check a few shaded squares against the times table.
• Describing a pattern only as ‘it goes across’. Say what changes from one number to the next.
• Thinking the first common multiple of 4 and 6 is 24. It is 12.

Answers

• Part A: the multiples of 9 to 100 (9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99) step one row down and one square left (adding 9 adds 10 and takes 1), so they lie on a diagonal to 90, with digit sums of 9; the multiples of 5 (20 of them) fill two columns, ending in 5 and in 0.
• Part B: 12, 24, 36, 48, 60, 72, 84, 96; they are the multiples of 12. Part C: 9: yes (the digit sum is a multiple of 9); 4: yes (the last two digits make a multiple of 4); 6: yes (it passes the tests for 2 and for 3); 144 ÷ 9 = 16, digit sum 9.

Multiples make patterns

A. Shade and describe

On Square A, start at 9 and keep adding 9, shading as you go. On Square B, do the same with 5.

The pattern for 9:

The pattern for 5:

B. Where two patterns meet

On Square C, ring the multiples of 4 in one colour and the multiples of 6 in another.

The numbers in both colours: _________________________________________________

What do they have in common? ___________________________________________

C. Predict, then test

Is 144 a multiple of 9, 4 or 6? Predict, then check with the quick tests.

A multiple of 9?   My prediction: yes / no    The test: __________________   yes / no

A multiple of 4?   My prediction: yes / no    The test: __________________   yes / no

A multiple of 6?   My prediction: yes / no    The test: __________________   yes / no

Square D is yours: choose a number, shade its multiples, and describe the pattern on the back.

Patterns that emerge

Children run algorithms that build sequences, by hand and on a calculator, and look hard at the results. Patterns emerge: adding 7 reaches every last digit before repeating, doubling gives the last digits 2, 4, 8, 6 over and over, and only square numbers have an odd number of factors.

We are learning to

• run an algorithm that builds a sequence, and record every result,
• identify and describe a pattern in the results,
• test whether a pattern keeps working.

Success criteria

• I can describe the pattern in the last digits when I keep adding 7.
• I can test a pattern on a new number and say whether it still works.

You need

One calculator per pair (on many calculators, pressing = again repeats the last + 7). The worksheet, one per child.

Lesson flow (about 50 minutes)

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

Watch for these ideas

• Writing only some of the results, so the pattern stays hidden. Record every step.
• Describing one example instead of the pattern. Say what always happens.
• Trusting a pattern after three examples. Test it on a new number.

Answers

• Part A: 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70; last digits 0, 7, 4, 1, 8, 5, 2, 9, 6, 3, 0: every digit once, then the pattern starts again. Adding 5 gives only the last digits 0 and 5.
• Part B: 1, 2, 4, 8, 16, 32, 64, 128, 256; after 1 the last digits go 2, 4, 8, 6 and repeat; each number is a factor of every number after it. Part C: 1, 4, 9, 16, 25, the square numbers (one factor pairs with itself).

Patterns that emerge

A. Start at 0 and add 7 each step

Write the last digit of each number. What pattern do you see?

Now try adding 5. Do you reach every last digit? __________________________

B. Start at 1 and double each step

Describe the pattern in the last digits. Is each number a factor of the next one?

C. An odd number of factors

Use the factor finder from Day 3. Which numbers from 1 to 30 have an odd number of factors? What kind of numbers are they?

Flowchart cards

Cut out the cards (Day 3), one set per pair. Arrange them into an algorithm that decides whether a number is divisible by 6, then join them with the yes and no labels. The blank cards are for your own steps.

Teacher note: a decision card has two ways out, so it needs a yes label and a no label. One working order is on the Day 3 plan.

Hundred squares

Four hundred squares for shading multiples (Day 4). Label each square with the rule you used.

Teacher note: one sheet per child. The patterns are described on the Day 4 plan.

What we know: Number Patterns and Algorithms

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

1. Use the digit sum: is 4812 divisible by 3?   __________
2. Is 4812 divisible by 9?   __________
3. Circle the numbers that divide 1230 exactly:   2    3    4    5    9    10
4. Which digit can go in the box so that 7□4 is divisible by 9?   __________
5. List every factor of 48.   ______________________________
6. Write the common multiples of 6 and 8 up to 50.   __________
7. Start at 3 and add 4 each step. Write five numbers, then describe the pattern.
8. Mia says every multiple of 6 is also a multiple of 3. Is she right? Explain.

Answers and marking guide

Answers

1. Digit sum 15, so yes (4812 ÷ 3 = 1604)
2. Digit sum 15, so no
3. 2, 3, 5, 10
4. 7
5. 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 (pairs to 6 × 8, stop at 7)
6. 24, 48
7. 3, 7, 11, 15, 19: all odd, each one less than a multiple of 4
8. Yes: 6 = 2 × 3, so any multiple of 6 is 3 times an even number.

A quick three-level guide

Eight questions, four ideas. A child at standard answers most questions and can say why, by naming the step, the test or the rule that decides.

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 a flowchart, quick tests, building an algorithm, shading and describing, and testing a pattern.