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

Five days of lessons for Year 4 Number. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet, plus cut-out algorithm cards and sequence and rule 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 two card sheets once; the cards are reused across the week.
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. An algorithm is a set of steps followed exactly, like a recipe. The ones this week start with a number, repeat a step that adds or multiplies, sometimes make a decision, and stop. The numbers they make show patterns, and describing those patterns is half the work.

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.

How an algorithm is written here

Every algorithm has three parts: a start (start at 3), a step to repeat (add 2), and a stop (stop after 6 numbers). Children write them as numbered steps or as a flowchart, where arrows show what to do next and a diamond holds a yes-or-no question.

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 is about following steps exactly. Days 2 and 3 give the two kinds of step the descriptor names, adding and multiplying, and Day 4 adds decisions. Day 5 turns it round: children start from the numbers, find the algorithm, and then create their own. It builds on skip counting and the times tables, and on odd and even numbers (AC9M4N02).

Materials for the week (one trip)

• From the classroom: scissors, pencils, coloured pencils (Day 2), this pack printed.
• Optional: counters, and a spreadsheet on the board to run an adding rule for a hundred steps.
• Cut out once, use all week: the algorithm cards and the sequence and rule cards in this pack.

Words for the word wall

algorithm, step, start, stop, rule, repeat, sequence, pattern, add, double, multiply, decision, even, odd, flowchart

Dear families

This week in maths, Year 4 follows and makes algorithms: sets of steps that make numbers, such as start at 3 and add 2 each time. We look for the patterns in the numbers they make.

Try this at home

• Take turns giving a rule: start at 4, add 5. Say the numbers together. What do you notice about the last digits?
• Try doubling: start at 1 and double each time. How many doubles until you pass 1000?
• Try a rule with a decision: if the number is even, halve it; if it is odd, add 1. Start at 20. Where do you end up?
• Talk about everyday algorithms, such as a recipe or brushing your teeth. What would go wrong if a step was skipped?

My algorithms this week

Fill one row a day. Write a rule, the numbers it made, and a pattern you noticed.

Printed from the free seegongsik Number Algorithms teaching pack · seegongsik.com/au/y4/number/AC9M4N09/pack

Follow the steps

An algorithm is a set of steps followed exactly, in order. Start at 3, add 2, and stop after 6 numbers makes 3, 5, 7, 9, 11, 13 every time, for everyone. Children follow algorithms written as numbered steps and as a flowchart, and find the slip when someone has not followed the steps exactly.

We are learning to

• follow the steps of an algorithm exactly, in order,
• read a flowchart, including a yes-or-no question,
• check someone else’s numbers against the steps.

Success criteria

• I can follow an algorithm and write the numbers it makes.
• I can find the place where someone did not follow the steps.

You need

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

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with Ari’s numbers.

Watch for these ideas

• Skipping or repeating a step, as Ari did. Point to each step as you do it.
• Counting the stop wrongly: after 6 numbers counts the start number as the first.
• Stop before you pass 120 lets 120 in. 130 would pass it, so it is not written.

Answers

• Part A: 1: 3, 5, 7, 9, 11, 13; 2: 50, 60, 70, 80, 90, 100, 110, 120; 3: 4, 8, 12, 16, 20, 24, 28, 32; 4: 1, 4, 7, 10, 13, 16, 19; 5: 2, 4, 8, 16, 32, 64.
• Part B: Ari skipped 20 and carried on from 15, so every number after it is 5 too big. The right numbers are 5, 10, 15, 20, 25, 30.

Follow the steps

A. Follow each algorithm. Write the numbers it makes.

B. Find the slip

Ari followed this algorithm: Start at 5. Add 5. Stop after 6 numbers. He wrote 5, 10, 15, 25, 30, 35. Where did he go wrong? Write the right numbers.

Adding rules

An adding rule adds the same amount at every step, so the numbers go up in equal jumps. Children run four adding rules and describe what they see: start at 7 and add 10 keeps a 7 in the ones place while the tens go up by one. On a hundred chart the numbers from an adding rule line up in columns or slanting lines.

We are learning to

• run an algorithm that adds the same amount each time,
• describe a pattern in the numbers, using the digits, odd and even, or the jumps,
• see an adding pattern on a hundred chart.

Success criteria

• I can make the numbers for an adding rule.
• I can describe a pattern exactly: what changes, by how much, and what stays the same.

You need

Coloured pencils. The algorithm cards (cut-out sheet 1). The worksheet, one per child.

Lesson flow (about 50 minutes)

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

Watch for these ideas

• Adding the step to the start every time (2, 5, 5, 5) instead of to the last number.
• Describing a pattern as “it gets bigger”. Ask what changes, by how much, and what stays the same.
• Finding add 9 hard. It is add 10, take 1, which is why the ones digit goes down by one each time.

Answers

• Part A: 2, 4, 6, 8, 10, 12 (all even, up by 2 each time); 5, 10, 15, 20, 25, 30 (the ones digit goes 5, 0, 5, 0); 7, 17, 27, 37, 47, 57 (the ones digit is always 7 and the tens go up by 1); 1, 10, 19, 28, 37, 46 (the tens go up by 1 and the ones go down by 1 (add 10, take 1)).
• Part B: 33 numbers, 3 to 99, every third square, making slanting lines. Part C: start at 4, add 5.

Adding rules

A. Run each rule for six numbers. What pattern do you notice?

Multiplying rules

A multiplying rule multiplies by the same number at every step. Doubling from 1 makes 1, 2, 4, 8, 16, 32, 64: the jumps get bigger each time, so a multiplying rule soon grows much faster than an adding rule. Children run multiplying rules, race one against an adding rule, and see how quickly doubling grows.

We are learning to

• run an algorithm that doubles or multiplies each time,
• describe how the numbers grow, including the jumps between them,
• compare a multiplying rule with an adding rule.

Success criteria

• I can make the numbers for a multiplying rule.
• I can explain why a multiplying rule overtakes an adding rule.

You need

A sheet of scrap paper for the opening. The worksheet, one per child. Calculators are optional, for checking.

Lesson flow (about 50 minutes)

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

Watch for these ideas

• Adding instead of multiplying: double 8 is 16, not 10.
• Thinking doubling and adding 2 are the same because both can begin 2, 4. The next numbers are 8 and 6.
• Losing count of zeros when multiplying by 10: 3, 30, 300, 3000 gains one zero each step.

Answers

• Part A: 1, 2, 4, 8, 16, 32, 64; 1, 3, 9, 27, 81, 243; 5, 10, 20, 40, 80, 160; 3, 30, 300, 3000.
• Part B: add 5 makes 5, 10, 15, 20, 25, 30; doubling makes 5, 10, 20, 40, 80, 160. Doubling is ahead from number 3 on, and ends at 160 against 30. Part C: 1, 2, 4, 8, 16, 32, 64, 128 cents, so 128 cents ($1.28) on day 8. The $10 plan pays $80 over 8 days, but on day 20 alone the doubling plan pays 524 288 cents ($5242.88).

Multiplying rules

A. Follow each multiplying rule. Write the numbers it makes.

B. The race: both rules start at 5

Which rule is ahead after six numbers? From which number is it ahead?

C. Pocket money: 1 cent on day 1, doubled every day

How many cents on day 8? How much is that in dollars and cents?

Decisions in the steps

Some algorithms make a decision at every step: if the number is even, halve it; if it is odd, add 1. The number itself decides which branch to take. Children run this rule from different starts and notice that every start they try ends at 1. Then they run a decision rule that only adds, and one whose decision is when to stop.

We are learning to

• follow an algorithm with a decision, choosing the branch from the number,
• check even or odd at every step, not just the first,
• describe a pattern across several starts.

Success criteria

• I can follow a rule with a decision and say which branch I took each time.
• I can describe a pattern I notice across several starts.

You need

The decision cards (cut-out sheet 1). The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with 12.

Watch for these ideas

• Deciding once and using that branch every time. The check happens at every step.
• Halving by taking away 2: half of 10 is 5, not 8.
• Going past 1. The stop says stop when you reach 1; without it the rule would go 1, 2, 1, 2 for ever.

Answers

• Part A: 6, 3, 4, 2, 1 (5 numbers); 10, 5, 6, 3, 4, 2, 1 (7 numbers); 7, 8, 4, 2, 1 (5 numbers); 9, 10, 5, 6, 3, 4, 2, 1 (8 numbers). Every start ends 4, 2, 1. From 12: 12, 6, 3, 4, 2, 1.
• Part B: 4, 7, 8, 11, 12, 15, 16, 19. The jumps go 3, 1, 3, 1, the numbers go even, odd, even, odd, and every two steps add 4. Part C: 1, 4, 7, 10; 2, 5, 8, 11, 14, 17, 20. A start ending in 0 stops at once.

Decisions in the steps

A. Rule: if the number is even, halve it. If it is odd, add 1. Stop when you reach 1.

What do you notice about how every start ends?

B. Rule: if the number is even, add 3. If it is odd, add 1. Start at 4. Stop after 8 numbers.

Describe the pattern:

C. Follow the flowchart from 1, then from 2

Spot and describe the pattern

Today the work runs backwards. Children start from a set of numbers, work out the algorithm that made it, and continue it: 3, 6, 12, 24 doubles, so 48 and 96 come next. Comparing each number with the one before shows whether the rule adds (equal jumps) or multiplies (jumps that grow). Then children write algorithms of their own for a partner.

We are learning to

• work out the rule behind a set of numbers and continue it,
• describe the pattern in words,
• create an algorithm, with or without a decision, for a partner to follow.

Success criteria

• I can find the rule that made a set of numbers and say what comes next.
• I can write an algorithm with a start, a step and a stop for a partner.

You need

The sequence and rule cards (cut-out sheet 2), one set per pair. The worksheet, one per child.

Lesson flow (about 50 minutes)

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

Watch for these ideas

• Looking at one jump only: 3 to 6 could be add 3 or double. Check the next jump too.
• Describing the numbers but not the rule. “They go up” is not yet a rule.
• Writing an algorithm with no stop, so it never ends. Every algorithm needs one.

Answers

• Part A: add 3, then 16, 19; double it, then 48, 96; add 25, then 125, 150; multiply by 3, then 162, 486; add 4, then 21, 25; double it, then 160, 320.
• Part B: for example, always ends in 5; all odd; the tens go up by 1. Part C: algorithms vary; check for a start, a step and a stop. Card match: 3, 6, 9, 12 is start at 3, add 3; 3, 6, 12, 24 is start at 3, double it; 2, 6, 10, 14 is start at 2, add 4; 2, 6, 18, 54 is start at 2, multiply by 3; 4, 40, 400, 4000 is start at 4, multiply by 10; 4, 14, 24, 34 is start at 4, add 10; 1, 10, 19, 28 is start at 1, add 9; 1, 6, 11, 16 is start at 1, add 5.

Spot and describe the pattern

A. What is the rule? What are the next two numbers?

B. Describe two patterns in 5, 15, 25, 35, 45

C. Write an algorithm for a partner

My partner’s numbers:

Did they match mine? If not, which step was unclear?

Algorithm cards

Cut out the sixteen cards, one set per pair. An algorithm needs a start card, a step card or a decision card, and a stop card. Pairs build one, run it, and swap with another pair.

Teacher note: some mixes never stop. Start at 3, add 2, stop when you reach 1 runs for ever, because adding never comes back down to 1. That is a good thing for children to discover and explain.

Sequence cards and rule cards

Cut out the eight sequence cards and the eight rule cards (Day 5), one set per pair. Match each sequence to the rule that made it. Each start is used twice, so look past the first numbers.

Sequence cards

Rule cards

Teacher note: the key is on the Day 5 plan.

What we know: Number Algorithms

Follow each algorithm exactly, and show your numbers.

1. Start at 6. Add 4. Stop after 5 numbers.   ______________________________
2. Start at 2. Double it. Stop after 5 numbers.   ______________________________
3. If the number is even, halve it. If it is odd, add 1. Start at 12. Stop when you reach 1.   ______________________________
4. 7, 11, 15, 19, ...   The rule is ______________. The next two numbers are ______ and ______.
5. 1, 4, 16, 64, ...   The rule is ______________. The next number is ______.
6. Describe a pattern in 8, 18, 28, 38, 48.   ______________________________
7. After 5 numbers, which is bigger: start at 3 and add 3, or start at 3 and double it? ______ Show why.
8. Write an algorithm that makes 10, 20, 30, 40, 50.   ______________________________

Answers and marking guide

Answers

1. 6, 10, 14, 18, 22.
2. 2, 4, 8, 16, 32.
3. 12, 6, 3, 4, 2, 1.
4. Add 4; 23, 27.
5. Multiply by 4; 256.
6. For example: every number ends in 8; they are all even; the tens go up by 1.
7. Doubling: 3, 6, 12, 24, 48 against 3, 6, 9, 12, 15, so 48 beats 15.
8. Start at 10. Add 10. Stop after 5 numbers. (Or start at 10, add 10, stop before you pass 50.)

A quick three-level guide

Eight questions, three ideas. A child at standard follows any rule from this week exactly, names the rule behind a set of numbers, and writes a working algorithm of their own.

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, adding rules, multiplying rules, decisions, and patterns.