Rules That Grow 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 pattern and rule cards, position 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 pattern and rule cards (Days 2 and 3) and the position and rule cards (Day 4) 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: a pattern is made by a rule, and knowing the rule lets you go as far as you like. A step rule says how to get from one term to the next, such as add 3 or add a half. A position rule jumps straight to any term, such as term = 3 × position + 1. Some rules add the same amount each time; others multiply, and those grow much faster.

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 term is one number in a pattern; its position says where it sits (first, second, third). The step is the amount added each time. In this pack the stage of a shape pattern and the position of a term mean the same thing.

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 grows shapes and records them as numbers; Day 2 finds the step in number patterns; Day 3 lets the step be a fraction or a decimal; Day 4 turns a step rule into a position rule that finds any term at once; and Day 5 sets adding rules against multiplying ones. It builds on the Year 5 work of following and creating number patterns, and it prepares Year 7, where a position rule is written with a letter, such as 3n + 1, and graphed.

Materials for the week (one trip)

• From the classroom: scissors, pencils, this pack printed.
• For Day 1: matchsticks, pop sticks or straws, and square tiles or cubes, about 40 of each per pair (the worksheet has drawings too).
• A calculator for each pair on Day 4, for checking terms in Term race.
• Cut out once, use all week: the pattern and rule cards and the position and rule cards, one set per pair.

Words for the word wall

pattern, term, position, stage, step, rule, step rule, position rule, continue, predict, check, add, subtract, double, multiply

Dear families

This week in maths, Year 6 finds the rules behind growing patterns. We build patterns with sticks and tiles, find the step from one term to the next (including steps like a half or 0.25), and use a position rule to jump straight to the 100th term.

Try this at home

• Count in halves or quarters together: 2, 2.5, 3, 3.5, ... What is the 10th number?
• Make a pattern with spoons or coins that grows by the same number each time. Ask your child the rule.
• Save $2.50 a week on paper. In which week does the total pass $20?
• Compare adding 5 each day with doubling each day, starting from 5. Which wins after a week?

My patterns this week

Fill one row a day. Write a pattern you found or made, its rule and the 10th term.

Printed from the free seegongsik Rules That Grow Patterns teaching pack · seegongsik.com/au/y6/algebra/AC9M6A01/pack

Growing shapes

A growing shape pattern is made by a rule: each stage adds the same group. Children build matchstick and tile patterns, record the counts in a table, find the step, and work out how many pieces a far-off stage needs without building it.

We are learning to

• build and continue a growing shape pattern,
• record a shape pattern in a table and find the step,
• use a rule to say how many pieces any stage needs.

Success criteria

• I can say how many matchsticks stage 10 of the squares needs, and how I know.
• I can explain where the ‘plus 1’ comes from in the matchstick rule.

You need

Matchsticks or straws and square tiles, about 40 of each per pair, and the worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B by building the tile L.

Watch for these ideas

• Counting every piece again at each stage instead of using the step.
• Multiplying by the step and forgetting the start: stage 10 of the squares is not 3 × 10 = 30 sticks.
• Doubling stage 5 to get stage 10: the squares need 31 sticks, not 2 × 16.

Answers

• Part A: 4, 7, 10, 13, 16; the step is 3; stage 10 needs 31; sticks = 3 × stage + 1 (one upright stick to start, then 3 for each square).
• Part B: 3, 5, 7, 9, 11; the step is 2; stage 10 needs 21; tiles = 2 × stage + 1 (the corner tile, then one more on each arm).
• Part C: any pattern that adds 4 a stage. A plus sign that grows one tile on each arm has 5, 9, 13, ... tiles, so stage 10 has 41.

Growing shapes

A. Matchstick squares

The step is ______. Stage 10 needs ______ sticks. Rule: sticks = ______ × stage + ______

B. A growing L

The step is ______. Stage 10 needs ______ tiles. Rule: tiles = ______ × stage + ______

C. Your own pattern

Design a pattern that grows by 4 pieces each stage. Draw stages 1, 2 and 3, then write the rule.

Rule: pieces = ______ × stage + ______    Stage 10 needs ______ pieces.

The step between terms

A step rule says what to add or subtract to get from one term to the next. Children find the step of number patterns that rise and fall, continue them, fill in missing terms, and decide whether a number belongs in a pattern without listing every term.

We are learning to

• find the constant step of a number pattern, rising or falling,
• continue a pattern and fill in missing terms,
• decide whether a number belongs in a pattern without listing every term.

Success criteria

• I can find the missing terms in 12, __, 26, 33, __.
• I can explain whether 74 is in the pattern that starts at 2 and adds 12.

You need

The worksheet, one per child, and the pattern and rule cards marked Day 2 (cut-outs), one set per pair.

Lesson flow (about 50 minutes)

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

Watch for these ideas

• Taking the step from the first two terms only, when the pattern doubles.
• Reading a falling pattern as rising: 50, 44, 38 subtracts 6.
• Sharing a gap among the wrong number of jumps: two missing terms make three jumps, not two.

Answers

• Part A: add 5, then 24, 29; subtract 6, then 26, 20; add 11, then 51, 62; subtract 13, then 48, 35; add 25, then 225, 250.
• Part B: 19 and 40 (add 7); 37 and 19 (subtract 6); 10 and 17 (add 7).
• Part C: 74 yes, term 7, 100 no, it falls between 98 and 110; 10 yes, term 11, 0 no, it falls between 2 and −6.

The step between terms

A. Find the step, then continue

B. Fill the gaps

Find the step first. Then write the missing terms.

1. 12 2633   step: ______

2.  3125 13  step: ______

3. 3  24  step: ______

C. Is it in the pattern?

Do not list every term. Work out how many jumps it would take.

1. Start at 2 and add 12 each time. Is 74 in the pattern? Is 100? How do you know?
____________________________________________________________________
____________________________________________________________________

2. Start at 90 and subtract 8 each time. Is 10 in the pattern? Is 0? How do you know?
____________________________________________________________________
____________________________________________________________________

Fraction and decimal steps

A step does not have to be a whole number. Patterns can climb by a half, a quarter or 0.05, or fall by 0.2, and the same pattern can be written in fractions or in decimals. Children continue patterns with these steps, write them in both forms, and solve a savings problem.

We are learning to

• continue patterns that step up or down by fractions or decimals,
• write a pattern in fractions and in decimals,
• use a fraction or decimal step to answer a question about money.

Success criteria

• I can continue 1/3, 2/3, 1, 1 1/3, ... and say the step.
• I can show that 0.25, 0.5, 0.75 and 1/4, 1/2, 3/4 are the same pattern.

You need

The worksheet, one per child, and all the pattern and rule cards, with the Day 3 cards added.

Lesson flow (about 50 minutes)

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

Watch for these ideas

• Counting decimals like whole numbers: after 0.9 comes 1, not 0.10.
• Adding tops and bottoms: 1/4 + 1/4 is 2/4, a half, not 2/8.
• Reading 3/2 as 3.2: three halves is 1.5.

Answers

• Part A: add 0.5: 2.5, 3; subtract 0.2: 1.6, 1.4; add 0.05: 0.25, 0.3; add 0.25: 2.25, 2.5. The 10th term of the talk prompt is 0.6.
• Part B: add 1/4: 1 1/4, 1 1/2; add 1/3: 1 2/3, 2; subtract 1/2: 3, 2 1/2; add 3/5: 2 4/5, 3 2/5.
• Part C: 1/4, 1/2, 3/4, 1; 0.2, 0.4, 0.6, 0.8; 1 1/2, 3, 4 1/2, 6. Part D: week 8, as 8 × $2.50 = $20.

Fraction and decimal steps

A. Decimal steps

B. Fraction steps

C. The other form

Write each pattern again: decimals as fractions, fractions as decimals.

1. 0.25, 0.5, 0.75, 1   is   ______________________________

2. 1/5, 2/5, 3/5, 4/5   is   ______________________________

3. 1.5, 3, 4.5, 6   is   ______________________________

D. The savings jar

Ari puts $2.50 in a jar at the end of each week. Write the total after each of the first four weeks. In which week does the total reach $20?

______, ______, ______, ______    Week ______

Position rules

A step rule walks from term to term; a position rule jumps straight to any term. The rule term = 4 × position gives the 25th term as 100 without listing the others. Children use position rules, turn step rules into position rules and work backwards from a term to its position.

We are learning to

• use a position rule to find any term,
• turn a step rule into a position rule,
• use a position rule to work backwards from a term to its position.

Success criteria

• I can find the 20th term of 5, 8, 11, 14, ... without listing the terms.
• I can say which stage of the matchstick squares uses 100 sticks.

You need

The worksheet, one per child, a calculator for each pair, and the position and rule 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 first pattern in Part B.

Watch for these ideas

• Using the step as the whole rule: 5, 8, 11 is not term = 3 × position.
• Adding the step to the position instead of multiplying by it.
• Undoing in the wrong order: to undo × 3 then + 1, take away 1 first, then divide by 3.

Answers

• Part A: term = 4 × position: 4, 8, 12, 16, 20, 40; term = 3 × position + 2: 5, 8, 11, 14, 17, 32; term = 0.5 × position: 0.5, 1, 1.5, 2, 2.5, 5; term = 10 × position − 1: 9, 19, 29, 39, 49, 99.
• Part B: step 3, term = 3 × position + 2, the 20th term 62; step 5, term = 5 × position + 1, the 20th term 101; step 2, term = 2 × position + 5, the 20th term 45; step 0.5, term = 0.5 × position + 1, the 20th term 11. In each, the step multiplies and the first term minus the step is added (for the first, 5 − 3 = 2).
• Part C: stage 50 needs 151 sticks; 100 sticks is stage 33; no stage uses 50 tiles, as 50 − 1 = 49 does not share into 2s (the counts are all odd).

Position rules

A. Use the rule

B. From a step rule to a position rule

The step is the number you multiply by. Then ask: what do you add to make the first term?

C. Back to the shapes

1. The matchstick squares from Day 1 use sticks = 3 × stage + 1. How many sticks does stage 50 need? ____________ Which stage uses 100 sticks? ____________

2. The growing L uses tiles = 2 × stage + 1. Can a stage use exactly 50 tiles? Explain. ________________________________________

D. Term race score sheet

Write the rule card and the position card you turned over, then the term. Check it on a calculator.

Add or multiply?

Some rules add the same amount each time; others multiply by the same amount. Adding grows in even steps, but multiplying grows faster and faster. Children sort patterns by their kind of rule, race an adding rule against a doubling rule, and see that two terms are not always enough to know the rule.

We are learning to

• tell a rule that adds from a rule that multiplies,
• compare how fast adding and multiplying rules grow,
• predict a term and check it against the rule.

Success criteria

• I can say why 3, 6, 12, 24 is not an adding pattern.
• I can find the stage where doubling from 5 passes adding 10 from 5.

You need

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

Lesson flow (about 50 minutes)

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

Watch for these ideas

• Taking the step from the first two terms when the pattern multiplies: 3, 6, 12 is not add 3.
• Thinking a multiplying pattern must get bigger: halving 64 gives 32, 16, 8.
• Thinking the rule that starts ahead stays ahead.

Answers

• Part A: multiplies, double: 48, 96; adds, add 3: 15, 18; multiplies, multiply by 10: 50 000, 500 000; multiplies, halve: 4, 2; adds, subtract 8: 48, 40; multiplies, multiply by 3: 8.1, 24.3; adds, add 1.5: 7.5, 9.
• Part B: adding 5, 15, 25, 35, 45, 55, 65; doubling 5, 10, 20, 40, 80, 160, 320. Doubling passes adding at stage 4 (40 against 35) and stays ahead.
• Part C: add 2: 6, 8, 10; double: 8, 16, 32. A third term tells them apart.

Add or multiply?

A. Sort the rules

B. The race

Pattern A starts at 5 and adds 10. Pattern B starts at 5 and doubles. Fill in the table.

B passes A at stage ______. Will A ever catch up again? ________________________

C. One start, two rules

A pattern begins 2, 4, ... Write two different rules that fit, and the next three terms for each.

Rule 1: ______________   next: ______, ______, ______

Rule 2: ______________   next: ______, ______, ______

Pattern cards and rule cards

Cut out the twelve pattern cards and the twelve rule cards once, one set for each pair.

Rule match (Days 2 and 3)

1. Use the cards for today and the days before. Lay the rule cards face up; shuffle the pattern cards face down.
2. Take turns to turn over a pattern card and point to its rule card. Say the next term to keep the pair.
3. If the rule is wrong, the pattern card goes to the bottom of the pile.
4. When the pile is gone, the player with more pairs wins.

Teacher note: the pairs are 2, 4, 6, 8 with start at 2, add 2; 2, 4, 8, 16 with start at 2, double; 4, 6, 8, 10 with start at 4, add 2; 3, 7, 11, 15 with start at 3, add 4; 15, 11, 7, 3 with start at 15, subtract 4; 7, 11, 15, 19 with start at 7, add 4; 0.5, 1, 1.5, 2 with start at 0.5, add 0.5; 1/4, 1/2, 3/4, 1 with start at 1/4, add 1/4; 2.5, 2.3, 2.1, 1.9 with start at 2.5, subtract 0.2; 1/2, 1, 2, 4 with start at 1/2, double; 1.2, 1.5, 1.8, 2.1 with start at 1.2, add 0.3; 1/3, 1, 1 2/3, 2 1/3 with start at 1/3, add 2/3. The cards come in near-miss families: the same start with a different step, and the same first two terms with add or double.

Position cards and rule cards

Cut out the eight rule cards and the eight position cards once, one set for each pair.

Term race (Day 4)

1. Shuffle the rule cards and the position cards into two piles, face down.
2. Turn over one card from each pile. Both players work out the term on paper.
3. Check on a calculator. The first correct answer wins the position card; the rule card goes to the bottom.
4. After eight rounds, the player with more position cards wins.

Teacher note: every term the cards can make.

What we know: Rules That Grow Patterns

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

1. A row of matchstick triangles uses 3, 5, 7, 9 sticks for stages 1 to 4. How many sticks does stage 10 use? ______________
2. Continue the pattern: 8, 15, 22, 29, ______, ______
3. Fill the gap: 45, ______, 31, 24
4. Continue the pattern: 1.2, 1.5, 1.8, ______, ______
5. Continue the pattern: 3/4, 1 1/2, 2 1/4, ______, ______
6. The rule is term = 5 × position − 2. Write the terms at positions 1, 2, 3 and 20. ______________________________
7. Write a position rule for 4, 7, 10, 13, ... and find the 30th term. ______________________________
8. Does 2, 6, 18, 54, ... add or multiply? What comes next? ______________________________

Answers and marking guide

Answers

1. 21 (add 2 a stage: sticks = 2 × stage + 1).
2. 36, 43.
3. 38 (subtract 7).
4. 2.1, 2.4.
5. 3, 3 3/4.
6. 3, 8, 13, 98.
7. term = 3 × position + 1; the 30th term is 91.
8. It multiplies by 3: next is 162.

A quick three-level guide

Eight questions, four ideas. A child at standard answers most questions and can say the rule that makes each pattern.

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: growing shapes, step rules, fraction and decimal steps, position rules, and adding against multiplying.