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

Five days of lessons for Year 3 Number. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet, plus algorithm cards, a hundred chart, 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 algorithm cards once; they are reused all week. Print a hundred chart for each child for Day 4.
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. An algorithm is a list of steps carried out in order, like a recipe. In number work it is usually a start number and a rule (start at 2, add 3), sometimes with a decision (if the number is even, halve it) and a rule for when to stop. Running the steps makes patterns appear, and describing those patterns is the point of the week.

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.

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 to 3 add one part of an algorithm at a time: the steps, then a decision, then a rule for when to stop. Day 4 turns to what the steps make, the patterns in the numbers and the rule behind a sequence. Day 5 puts the parts together as children create their own algorithms. It leans on Multiply and Divide (AC9M3N04) and Times Tables and Their Division (AC9M3A03).

Materials for the week (one trip)

• From the classroom: scissors, pencils, coloured pencils, this pack printed.
• Counters for checking even and odd on Day 2, and a 0 to 50 number line on the board or floor.
• One or two copies of the hundred chart for each child (cut-out sheet 2).
• Cut out once, use all week: the algorithm cards in this pack.

Words for the word wall

algorithm, step, start, rule, sequence, decision, even, odd, halve, double, stop, pattern, repeat, last digit, create

Dear families

This week in maths, Year 3 is learning about algorithms: lists of steps carried out in order. Children follow number rules, make their own, and describe the patterns the rules make.

Try this at home

• Give your child a rule to follow aloud: start at 3 and add 4, five times. What numbers did you say?
• Play “guess my rule”: say 5, 10, 20, 40 and ask for the rule (double each time).
• Count in fives or twos together and listen for the pattern in the last digit.
• Spot algorithms in daily life: a recipe, the steps for brushing teeth, a set of directions.

Our rules this week

Fill one row a day: a rule you followed or made, and the first numbers it gave.

Printed from the free seegongsik Number Algorithms and Patterns teaching pack · seegongsik.com/au/y3/number/AC9M3N07/pack

Follow the steps

An algorithm is a list of steps carried out in order, like a recipe. The simplest number algorithm is a start and a rule: start at 2, add 3, and keep adding 3 to get 2, 5, 8, 11, 14. The start and the step decide the whole sequence.

We are learning to

• follow an algorithm with a start number and a step, one step at a time,
• record the numbers it makes as a sequence,
• check a sequence by following the steps again.

Success criteria

• I can follow a start-and-step algorithm.
• I can write the sequence it makes.

You need

A 0 to 50 number line on the board or floor. The algorithm cards (cut-out sheet 1), one set per pair. The worksheet.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after the board; start Session B with the worksheet.

Watch for these ideas

• Adding the step to the wrong number (2, 5, 8, 10). Each new number comes from the one just before it.
• Counting the start as a jump (start at 2, add 3: 3, 6, 9). The start is the first number.
• Slipping when the numbers get bigger. Say each step aloud: 14 add 3 is 17.

Answers

• Start at 4. Add 2 each time: 4, 6, 8, 10, 12, 14.
• Start at 1. Add 5 each time: 1, 6, 11, 16, 21, 26.
• Start at 10. Add 10 each time: 10, 20, 30, 40, 50, 60.
• Start at 30. Take away 4 each time: 30, 26, 22, 18, 14, 10.
• Start at 7. Add 4 each time: 7, 11, 15, 19, 23, 27.
• Ari: 14 is the first slip (11 add 4 is 15). The sequence is 3, 7, 11, 15, 19, 23.

Follow the steps

Follow each algorithm. The first box is the start. Write the next five numbers.

Find the slip

Ari followed “Start at 3. Add 4 each time.” He wrote:

Circle the first wrong number. Then write the sequence correctly:

A step with a decision

Some algorithms check the number at every step and choose what to do. The rule “if the number is even, halve it; if it is odd, add 1” sends 12 to 6, then 3, then 4, 2 and 1, where it stops. Following it means checking the number every time.

We are learning to

• tell even and odd numbers apart (even numbers end in 0, 2, 4, 6 or 8),
• follow an algorithm with a decision, checking the number at every step,
• record the path from the start to the stop.

Success criteria

• I can follow an algorithm that makes a decision.
• I can say which way each number goes, and why.

You need

Counters, about 20 per pair. The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after the board; start Session B with the worksheet and Longest path.

Watch for these ideas

• Halving an odd number (3 becomes 1 and a half). Check even or odd first.
• Thinking a number ending in 0 is odd. Numbers ending in 0, 2, 4, 6 or 8 are even.
• Forgetting to stop at 1. Checking for the stop is a step of the algorithm too.

Answers

• Even: 14, 30, 8, 46.
• Paths: 10, 5, 6, 3, 4, 2, 1; 7, 8, 4, 2, 1; 16, 8, 4, 2, 1; 9, 10, 5, 6, 3, 4, 2, 1.
• Notice: every path ends 4, 2, 1. Of the four, 9 takes the most steps (7). From 2 to 12, 9 is the longest too.

A step with a decision

Even or odd?

Circle the even numbers.

The algorithm

Follow it

The first box is the start. You may not need every box.

What do you notice?

Which numbers are in every path? Which start took the most steps?

Stop when told

Many algorithms need a rule for when to stop, or they would run for ever. “Double it; stop once it is past 20” runs 3, 6, 12, 24 and halts, because 24 is the first number past 20. The stop rule is checked after every step.

We are learning to

• follow an algorithm with a stop rule,
• check the stop rule after every step and stop at the right moment,
• compare stop rules: past a number, less than a number, at a number, or after a count.

Success criteria

• I can follow an algorithm to its stop.
• I can say why it stopped where it did.

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 the board; start Session B with the worksheet and the cards.

Watch for these ideas

• Stopping before the limit (at 12, because the next is too big). Past 20 means 24 is written.
• Running past the stop (3, 6, 12, 24, 48). Check the stop rule after every step.
• Mixing up “past 20” and “at 20”. Past 20 means more than 20.

Answers

• Clap until: 16 (the first number past 15), then 10 (after 5 claps).
• 1. 5, 10, 20, 40, 80; 2. 4, 10, 16, 22, 28, 34; 3. 50, 43, 36, 29, 22, 15, 8; 4. 1, 4, 7, 10, 13, 16; 5. 2, 4, 8, 16, 32, 64.
• Most numbers: algorithm 3, with seven. New stop for 1: “Stop when you reach 40” or “Stop after 4 numbers”.

Stop when told

The first box is the start. Check the stop rule after every number. You may not need every box.

Which made the most numbers?

Algorithm ______ made the most numbers. It made ______ numbers.

Change a stop rule

Change the stop rule of algorithm 1 so that it stops at 40. Write your new stop rule.

Patterns that appear

Running a rule again and again makes patterns appear: count in fives and the last digit goes 5, 0, 5, 0. Describing a pattern also works backwards. Look at 3, 6, 12, 24 and name the rule that made it: double each time.

We are learning to

• run a counting algorithm and look for a pattern in the last digits,
• describe a pattern in words (the last digit goes 5, 0, 5, 0),
• name the rule behind a sequence: add or take away the same number, double or halve.

Success criteria

• I can describe a pattern that appears.
• I can name the rule behind a sequence.

You need

The hundred chart (cut-out sheet 2), one per child, and two coloured pencils. The worksheet.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after The pattern emerges; start Session B with Name the pattern.

Watch for these ideas

• Listing the numbers (5, 10, 15, 20) instead of the pattern. Ask what keeps happening.
• Calling every growing sequence adding. If the jumps grow (3, 6, 12), it is doubling.
• Naming the rule from two numbers (3 then 6 could be add 3 or double). Check it on all of them.

Answers

• Count in twos from 2: 2, 4, 6, 8, 10, 12, 14, 16; Count in fours from 4: 4, 8, 12, 16, 20, 24, 28, 32; Count in tens from 3: 3, 13, 23, 33, 43, 53, 63, 73.
• Last digits: twos go 2, 4, 6, 8, 0 and repeat; fours go 4, 8, 2, 6, 0 and repeat; tens from 3 always end in 3.
• Rules: add 4 (20, 24); double (80, 160); take away 5 (30, 25); halve (4, 2); add 5 (26, 31).
• Hundred chart: the fives make two columns, ending in 5 and in 0. The numbers coloured twice all end in 0.

Patterns that appear

Last-digit patterns

Finish each count. Then write the last digits and describe the pattern.

Name the rule

Write the next two numbers and the rule.

4, 8, 12, 16, ____, ____    The rule: ______________________

5, 10, 20, 40, ____, ____    The rule: ______________________

50, 45, 40, 35, ____, ____    The rule: ______________________

64, 32, 16, 8, ____, ____    The rule: ______________________

6, 11, 16, 21, ____, ____    The rule: ______________________

Build your own rule

The last step is to create an algorithm: choose a start, a step and a stop, run it, and describe the pattern it makes. Start at 4 and add 5 to get 4, 9, 14, 19, 24: the last digit goes 4, 9, 4, 9.

We are learning to

• create an algorithm by choosing a start, a step and a stop,
• run it and record the sequence it makes,
• describe its pattern, and read a partner’s rule from their numbers.

Success criteria

• I can create an algorithm with a start, a step and a stop.
• I can describe the pattern my algorithm makes.

You need

The algorithm cards (cut-out sheet 1) for ideas. The worksheet, one per child. The unit’s self-check quiz.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after My algorithm 1; start Session B with algorithm 2 and the quiz.

Watch for these ideas

• Making a rule with no stop. Every algorithm on the sheet needs a stop rule.
• Describing the rule (add 5) instead of the pattern. The pattern is what keeps happening in the numbers.
• Giving a partner only two numbers (2, 4 could be add 2 or double). Give at least four.

Answers

• Opening: 10, 12, 14, 16, 18; 10, 15, 20, 25, 30; 10, 20, 30, 40, 50. The step sets the jump; the last digits change with it.
• Own algorithms: any start, step and stop run correctly. Check the stop is reached and the pattern is about the numbers.
• With a decision: for example “if even, add 3; if odd, add 1” from 2 gives 2, 5, 6, 9, 10, 13.

Build your own rule

Choose a start, a step and a stop. Run your algorithm, then describe the pattern it makes.

Partner check

Write four of your numbers for a partner. They write the rule they think you used.

My partner thinks my rule is: ____________________________   Right? yes / no

What I learned

Algorithm cards

Cut out the sixteen cards and sort them into three piles: start, step and stop. Pairs draw one card from each pile they need, run the algorithm, and write the numbers it makes.

Teacher note: Day 1 uses start and step cards only; Days 3 and 5 add a stop card. Every step makes the numbers grow, so every combination reaches its stop.

Hundred chart

One or two copies per child. On Day 4, colour each number you land on when you count in fives, then use a second colour for the count in twos.

Teacher note: the fives fill two columns (ending in 5 and 0); the twos fill every second column; both colours meet on the numbers ending in 0.

What we know: Number Algorithms and Patterns

Work on your own.

1. Start at 3. Add 4 each time. Write the first five numbers: ______________________________
2. Circle the even numbers:   12    25    40    37    68
3. Rule: stop at 1; if even, halve it; if odd, add 1. Start at 6 and write the path: ______________________________
4. Start at 4. Double it. Stop once it is past 30. Write the numbers: ______________________________
5. Count in twos from 10: 10, 12, 14, 16, 18, 20. What pattern do the last digits make? ______________________________
6. What is the rule?   9, 12, 15, 18    ______________________________
7. What is the rule?   6, 12, 24, 48    ______________________________
8. Make an algorithm that gives 5, 15, 25, 35. Start: ______ Step: ______________

Answers and marking guide

Answers

1. 3, 7, 11, 15, 19.
2. 12, 40 and 68.
3. 6, 3, 4, 2, 1.
4. 4, 8, 16, 32 (32 is the first number past 30).
5. 0, 2, 4, 6, 8, and then it starts again.
6. Add 3 each time.
7. Double each time.
8. Start at 5, add 10 each time.

A quick three-level guide

Eight questions, three ideas. A child at standard follows a rule with a decision and a stop, and can say in words what keeps happening in a sequence.

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, decisions, stop rules, patterns and rules, and creating algorithms.