C. Algebra · Patterns, Expressions, Equations, Inequalities, Coding and ModellingPattern Rules: Predictions, Missing Elements and Unknown Values
C1.3 A1About 40 minutes
Learning goals- I can use a pattern rule to extend a pattern and to predict terms far ahead.
- I can justify my predictions using the pattern.
- I can find missing elements in repeating, growing and shrinking patterns.
- I can use the equation of a linear growing pattern to find which term has a certain value.
Words to know
pattern rule
A rule that tells how a pattern works, in words or as an equation like t = 4n + 2.
extend
Continue a pattern by finding the next terms.
predict
Use the rule to say what a term will be without building every term.
justify
Explain why your answer is right, using the pattern or the picture.
unknown value
A number we don't know yet, shown by a variable, like the n in 4n + 2 = 50.
missing element
A term or item left out of a pattern that you need to find.
Remember from Grade 5
In Grade 5 you wrote term-number rules to predict far ahead, like the 25th term, and you found missing elements in patterns. You also used a core to predict an element in a repeating pattern.
In Grade 6 you will also run a pattern backward: if you know the term value, you can use the equation to find the term number.
Let's start with pattern blocks
Nadia puts hexagon tiles in a row, touching side by side. She counts the edges all the way around the outside: that's the perimeter, measured in hexagon sides.
The pattern is 6, 10, 14, ... The jump is 4, so it is linear. For n = 1, 4n = 4, and the perimeter is 6: that's 2 more.
Equation: t = 4n + 2, where n is the number of hexagons and t is the perimeter.
Justify it with the picture: each hexagon has 2 slanted sides on top and 2 slanted sides on the bottom that are always on the outside (4 per hexagon, so 4n). The upright sides are hidden where hexagons touch, except for 1 at each end (+ 2).
Two jobs for an equation
Job 1: Predict forward. You know the term number. Find the value.
What is the perimeter of a train of 30 hexagons? t = 4 × 30 + 2 = 120 + 2 = 122.
Job 2: Solve backward. You know the value. Find the term number.
Which train has a perimeter of 50? Write 4n + 2 = 50 and undo the steps in reverse order:
| Step | Think | Result |
|---|
| Start | 4n + 2 = 50 | Something times 4, plus 2, makes 50 |
| Undo the + 2 | 50 − 2 = 48 | 4n = 48 |
| Undo the × 4 | 48 ÷ 4 = 12 | n = 12 |
| Verify | 4 × 12 + 2 = 48 + 2 = 50 ✔ | The 12th train has perimeter 50 |
Worked exampleExample 1: Predict and justify
Use t = 4n + 2 to predict the perimeter of a train of 25 hexagons. Justify your answer.
- t = 4 × 25 + 2 = 100 + 2 = 102.
- Justify: 25 hexagons each have 4 outside slanted sides (100), plus the 2 upright sides at the ends (2).
The perimeter is 102 sides.
Worked exampleExample 2: Find the term number
A pattern has the equation t = 3n + 5. Which term has the value 65?
- Write the equation: 3n + 5 = 65.
- Undo + 5: 65 − 5 = 60, so 3n = 60.
- Undo × 3: 60 ÷ 3 = 20, so n = 20.
- Verify: 3 × 20 + 5 = 60 + 5 = 65 ✔
Term 20 has the value 65.
Worked exampleExample 3: Is it in the pattern?
Is 100 a term in the pattern t = 3n + 2?
- Write 3n + 2 = 100. Undo + 2: 3n = 98.
- Is 98 a multiple of 3? 3 × 32 = 96 and 3 × 33 = 99. No.
- So no whole-number term number gives 100.
No, 100 is not in the pattern. The pattern goes ..., 95, 98, 101, ... and skips 100.
Worked exampleExample 4: Missing elements in a growing pattern
Find the missing terms: 7, ☐, 19, 25, ☐
- From 19 to 25 is one jump of +6.
- Check the gap from 7 to 19: that's 2 jumps. 19 − 7 = 12, and 12 ÷ 2 = 6 ✔. So the jump is +6 all the way.
- 7 + 6 = 13. 25 + 6 = 31.
Worked exampleExample 5: Missing elements in a shrinking pattern
Find the missing terms: 81, 27, ☐, 3, ☐
- 81 − 27 = 54. If it subtracted 54 each time, the next would be below zero. So try dividing.
- 81 ÷ 3 = 27 ✔. Then 27 ÷ 3 = 9, and 9 ÷ 3 = 3 ✔.
- 3 ÷ 3 = 1.
81, 27, 9, 3, 1: divide by 3 each time.
Worked exampleExample 6: Predict in a repeating pattern
The core is ❄️⛄🌲🌲. What is the 50th element? Justify.
- The core has 4 elements. Every 4th element ends a core: elements 4, 8, 12, ..., 48 are all 🌲 (the last one in the core).
- 50 ÷ 4 = 12 R2. So 12 full cores take us to element 48.
- Element 49 is the 1st of the next core (❄️), and element 50 is the 2nd (⛄).
Undo in reverse
To solve 4n + 2 = 50, think about how the value was built: multiply by 4, then add 2. To go backward, undo the last step first: subtract 2, then divide by 4.
Always verify by putting your answer back into the equation. If it doesn't work, that's useful information: go back and look for the slip.
Just for funFunny but true: In the ❄️⛄🌲🌲 pattern, the 1000th element is 🌲, because 1000 ÷ 4 = 250 with no remainder, so element 1000 ends a core. A pattern rule lets you predict much further than anyone could ever draw!
Hands-on activitySquare train perimeters
You need: 10 or more square things that are all the same size (sticky notes, coasters, square crackers or paper squares), paper and a pencil.
- Put squares in a row so they touch side to side. Count the outside edges (the perimeter, in square sides) for 1, 2, 3 and 4 squares.
- Make a table. Write an equation for the pattern.
- Predict: what is the perimeter of a train of 25 squares? Justify it with the picture.
- Solve backward: which train has a perimeter of 40? Build it to check if you have enough squares, or check with your equation.
- Can any train have a perimeter of 31? Why or why not?
What to look for: 4, 6, 8, 10; t = 2n + 2; 25 squares → 52 (each square has a top and a bottom edge outside, plus one edge at each end); 2n + 2 = 40 gives n = 19; 31 is impossible because every perimeter in this pattern is even.
Your turn
1Use t = 4n + 2 to find the perimeter of a train of 15 hexagons.
4 × 15 + 2 = 60 + 2 = 62.
2For t = 6n − 1, which term has the value 59?
6n − 1 = 59. Undo − 1: 6n = 60. Undo × 6: n = 10. Verify: 60 − 1 = 59 ✔.
3Find the missing terms: 4, 11, ☐, 25, ☐
The jump is +7 (4 + 7 = 11). 4, 11, 18, 25, 32.
4Find the missing terms: 96, 48, ☐, 12, ☐
Divide by 2 each time. 96, 48, 24, 12, 6.
5The core is 🍁🍁❄️. What is the 20th element?
20 ÷ 3 = 6 R2. After 6 full cores (18 elements), element 19 is the 1st and element 20 is the 2nd: 🍁.
6Is 100 a term in t = 4n + 4?
4n + 4 = 100, so 4n = 96 and n = 24. Yes, it is term 24. Verify: 96 + 4 = 100 ✔.
7Which hexagon train has a perimeter of 82?
4n + 2 = 82. 4n = 80, so n = 20. The train of 20 hexagons.
More practice: warm-up
1Use t = 2n + 1 to find t when n = 6.
2What comes next: 3, 8, 13, 18, ...?
3Find the missing term: 10, 20, ☐, 40
n = 7, because 5 × 7 = 35.
5The core is ⭐🌙. What is the 9th element?
Odd-numbered elements are ⭐, so the 9th is ⭐.
More practice: on your own
1Use t = 5n + 4 to predict term 25.
5 × 25 + 4 = 125 + 4 = 129.
2For t = 8n − 3, which term has the value 77?
8n − 3 = 77. 8n = 80, so n = 10. Verify: 80 − 3 = 77 ✔.
3A table for a linear pattern is missing two values: 1 → ☐, 2 → 16, 3 → ☐, 4 → 30. Find them and write the equation.
From term 2 to term 4 is 2 jumps: 30 − 16 = 14, so each jump is 7. Term 1: 16 − 7 = 9. Term 3: 16 + 7 = 23. Equation: 7n + 2 (check: 7 × 2 + 2 = 16 ✔), so t = 7n + 2.
4Find the missing terms: 100, ☐, 86, 79, ☐
86 − 79 = 7, so subtract 7 each time. 100 − 7 = 93 (and 93 − 7 = 86 ✔). 79 − 7 = 72.
5Mia has $15. She saves $6 every week. The equation t = 6n + 15 gives her savings after n weeks. After how many weeks will she have $75?
6n + 15 = 75. 6n = 60, so n = 10 weeks. Verify: 6 × 10 + 15 = 75 ✔.
Challenge
1Explain why a hexagon train (t = 4n + 2) can never have a perimeter of 75. Which perimeters come just before and just after 75?
4n is always even, and even + 2 is even, so every perimeter is even. 75 is odd, so it can't happen. Just before: 4n + 2 = 74 gives n = 18. Just after: 4 × 19 + 2 = 78. So 74 and 78.
2The core is ❄️⛄🌲🌲. How many 🌲 are in the first 50 elements?
50 ÷ 4 = 12 R2. 12 full cores have 12 × 2 = 24 🌲. The 2 extra elements are ❄️⛄, with no trees. 24 trees.
3Pattern A is t = 3n + 4. Pattern B is 2, 7, 12, 17, ... Is there a term number where both patterns have the same value?
Pattern B's equation is t = 5n − 3. Compare: n = 3 gives A = 13, B = 12. n = 4 gives A = 16, B = 17. A is ahead at term 3 and B is ahead at term 4, and they can't be equal in between because there is no term 3½. No, they never have the same value.
Quiz
1. Use t = 3n + 4. What is term 20?
2. For t = 2n + 6, which term has the value 30?
3. Find the missing term: 5, ☐, 17, 23
4. A repeating pattern has the core A B C. What is the 30th element?
5. To solve 4n + 2 = 50, what should you do?
6. Is 45 a term in t = 5n + 3?
Big idea: Grade 6 students determine pattern rules and use them to extend patterns, make and justify predictions, and find missing elements in repeating, growing and shrinking patterns. They also use the equation of a linear growing pattern to solve for an unknown term number (C1.3). Solving 4n + 2 = 50 by undoing the steps (subtract 2, then divide by 4) is an early form of equation solving.
Common difficulties:
- Undoing in the wrong order (dividing 50 by 4 first). Ask: "What was done last when the value was built?"
- Not verifying. Encourage your child to substitute the answer back in every time.
- Off-by-one errors with cores. In a core of 4, element 50 is the 2nd element of a core (50 ÷ 4 = 12 R2), not the 50th of anything new.
How to help: Use sticky notes or crackers to build square trains and ask both kinds of questions: "How many for 25?" and "Which train has 40?" Ask your child to justify using the picture, not only the equation. Social-emotional learning (A1): a wrong first guess is part of the process; praise checking and trying again.