← A rule is only found once it survives every example · Generalize a Growing Pattern into a Rule

A rule is only found once it survives every example · 12 practice problems

6.EE.A.25.OA.B.36.NS.B.3

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 0.72

Find the rule and work out the number that belongs in the empty space.

11 8 0.76 20 12 1.28 11 3 0.56 8 10
Show solution
1 · Understandwhat's really being asked

Three pairs of numbers each have a result written under them, and the fourth pair's result is missing. We need the rule, and then the missing number.

Givens
  • 11, 8 gives 0.76.
  • 20, 12 gives 1.28.
  • 11, 3 gives 0.56.
  • The pair 8, 10 has no result yet.
Unknowns
  • The number under the last pair.
Constraints
  • The same rule applies to every group.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check#13 Convert to Algebra

Guess a rule from one group and it will usually fit that group alone. Test each guess against all three finished groups before using it, and start with the sums, which are easy to compare with the results.

3 · Execute4 carry out the plan

1Add each pair

#6 Guess and Check 5.OA.B.3
Adding is the first thing to try, and it makes the three results easy to line up.
11+8=19,20+12=32,11+3=1411 + 8 = 19 ,\quad 20 + 12 = 32 ,\quad 11 + 3 = 14
Three sums to compare against three results.

2Compare each sum with its result

#5 Look for a Pattern 6.NS.B.3
19 gives 0.76 and 32 gives 1.28; each result is the sum divided by the same number.
19÷0.76=25,32÷1.28=2519 \div 0.76 = 25,\quad 32 \div 1.28 = 25
Every group divides by 25.

3State the rule and test it

#13 Convert to Algebra 6.EE.A.2
The rule is (first + second) divided by 25. Checking it on the third group: 11 + 3 is 14, and 14 over 25 is 0.56 -- which is what is printed.
(□+□)÷25(\square + \square) \div 25
It survives all three, so it is the rule.

4Apply it to the last pair

#5 Look for a Pattern 6.NS.B.3
8 plus 10 is 18, divided by 25.
(8+10)÷25=0.72(8 + 10) \div 25 = 0.72
The blank is 0.72.
Answer: 0.72
4 · Reviewdoes it hold up?

Going backwards, 0.72 times 25 is 18, which is exactly 8 plus 10.

Another way: Subtracting or multiplying the pairs also gets tried, but 11 and 8 would then give 3 or 88, and neither is 0.76.

Standardsmin grade 6
  • 6.EE.A.2 Write, read, and evaluate expressions with variables — Writing the rule as an expression rather than describing it.
  • 5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Reading the pairs and results as a pattern.
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing a whole number to a decimal result.
💡Takeaway. A rule that fits one example is a guess. A rule that fits all of them is a rule.
Variant 2 easy answer: 6.75

Find the rule and work out the number that belongs in the empty space.

8 15 5.75 11 5 4 10 21 7.75 18 9
Show solution
1 · Understandwhat's really being asked

Three pairs of numbers each have a result written under them, and the fourth pair's result is missing. We need the rule, and then the missing number.

Givens
  • 8, 15 gives 5.75.
  • 11, 5 gives 4.
  • 10, 21 gives 7.75.
  • The pair 18, 9 has no result yet.
Unknowns
  • The number under the last pair.
Constraints
  • The same rule applies to every group.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check#13 Convert to Algebra

Guess a rule from one group and it will usually fit that group alone. Test each guess against all three finished groups before using it, and start with the sums, which are easy to compare with the results.

3 · Execute4 carry out the plan

1Add each pair

#6 Guess and Check 5.OA.B.3
Adding is the first thing to try, and it makes the three results easy to line up.
8+15=23,11+5=16,10+21=318 + 15 = 23 ,\quad 11 + 5 = 16 ,\quad 10 + 21 = 31
Three sums to compare against three results.

2Compare each sum with its result

#5 Look for a Pattern 6.NS.B.3
23 gives 5.75 and 16 gives 4; each result is the sum divided by the same number.
23÷5.75=4,16÷4=423 \div 5.75 = 4,\quad 16 \div 4 = 4
Every group divides by 4.

3State the rule and test it

#13 Convert to Algebra 6.EE.A.2
The rule is (first + second) divided by 4. Checking it on the third group: 10 + 21 is 31, and 31 over 4 is 7.75 -- which is what is printed.
(□+□)÷4(\square + \square) \div 4
It survives all three, so it is the rule.

4Apply it to the last pair

#5 Look for a Pattern 6.NS.B.3
18 plus 9 is 27, divided by 4.
(18+9)÷4=6.75(18 + 9) \div 4 = 6.75
The blank is 6.75.
Answer: 6.75
4 · Reviewdoes it hold up?

Going backwards, 6.75 times 4 is 27, which is exactly 18 plus 9.

Another way: Subtracting or multiplying the pairs also gets tried, but 8 and 15 would then give 7 or 120, and neither is 5.75.

Standardsmin grade 6
  • 6.EE.A.2 Write, read, and evaluate expressions with variables — Writing the rule as an expression rather than describing it.
  • 5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Reading the pairs and results as a pattern.
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing a whole number to a decimal result.
💡Takeaway. A rule that fits one example is a guess. A rule that fits all of them is a rule.
Variant 3 easy answer: 5

Find the rule and work out the number that belongs in the empty space.

16 13 5.8 10 17 5.4 13 23 7.2 10 15
Show solution
1 · Understandwhat's really being asked

Three pairs of numbers each have a result written under them, and the fourth pair's result is missing. We need the rule, and then the missing number.

Givens
  • 16, 13 gives 5.8.
  • 10, 17 gives 5.4.
  • 13, 23 gives 7.2.
  • The pair 10, 15 has no result yet.
Unknowns
  • The number under the last pair.
Constraints
  • The same rule applies to every group.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check#13 Convert to Algebra

Guess a rule from one group and it will usually fit that group alone. Test each guess against all three finished groups before using it, and start with the sums, which are easy to compare with the results.

3 · Execute4 carry out the plan

1Add each pair

#6 Guess and Check 5.OA.B.3
Adding is the first thing to try, and it makes the three results easy to line up.
16+13=29,10+17=27,13+23=3616 + 13 = 29 ,\quad 10 + 17 = 27 ,\quad 13 + 23 = 36
Three sums to compare against three results.

2Compare each sum with its result

#5 Look for a Pattern 6.NS.B.3
29 gives 5.8 and 27 gives 5.4; each result is the sum divided by the same number.
29÷5.8=5,27÷5.4=529 \div 5.8 = 5,\quad 27 \div 5.4 = 5
Every group divides by 5.

3State the rule and test it

#13 Convert to Algebra 6.EE.A.2
The rule is (first + second) divided by 5. Checking it on the third group: 13 + 23 is 36, and 36 over 5 is 7.2 -- which is what is printed.
(□+□)÷5(\square + \square) \div 5
It survives all three, so it is the rule.

4Apply it to the last pair

#5 Look for a Pattern 6.NS.B.3
10 plus 15 is 25, divided by 5.
(10+15)÷5=5(10 + 15) \div 5 = 5
The blank is 5.
Answer: 5
4 · Reviewdoes it hold up?

Going backwards, 5 times 5 is 25, which is exactly 10 plus 15.

Another way: Subtracting or multiplying the pairs also gets tried, but 16 and 13 would then give 3 or 208, and neither is 5.8.

Standardsmin grade 6
  • 6.EE.A.2 Write, read, and evaluate expressions with variables — Writing the rule as an expression rather than describing it.
  • 5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Reading the pairs and results as a pattern.
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing a whole number to a decimal result.
💡Takeaway. A rule that fits one example is a guess. A rule that fits all of them is a rule.
Variant 4 easy answer: 6.8

Find the rule and work out the number that belongs in the empty space.

4 8 2.4 12 7 3.8 21 22 8.6 9 25
Show solution
1 · Understandwhat's really being asked

Three pairs of numbers each have a result written under them, and the fourth pair's result is missing. We need the rule, and then the missing number.

Givens
  • 4, 8 gives 2.4.
  • 12, 7 gives 3.8.
  • 21, 22 gives 8.6.
  • The pair 9, 25 has no result yet.
Unknowns
  • The number under the last pair.
Constraints
  • The same rule applies to every group.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check#13 Convert to Algebra

Guess a rule from one group and it will usually fit that group alone. Test each guess against all three finished groups before using it, and start with the sums, which are easy to compare with the results.

3 · Execute4 carry out the plan

1Add each pair

#6 Guess and Check 5.OA.B.3
Adding is the first thing to try, and it makes the three results easy to line up.
4+8=12,12+7=19,21+22=434 + 8 = 12 ,\quad 12 + 7 = 19 ,\quad 21 + 22 = 43
Three sums to compare against three results.

2Compare each sum with its result

#5 Look for a Pattern 6.NS.B.3
12 gives 2.4 and 19 gives 3.8; each result is the sum divided by the same number.
12÷2.4=5,19÷3.8=512 \div 2.4 = 5,\quad 19 \div 3.8 = 5
Every group divides by 5.

3State the rule and test it

#13 Convert to Algebra 6.EE.A.2
The rule is (first + second) divided by 5. Checking it on the third group: 21 + 22 is 43, and 43 over 5 is 8.6 -- which is what is printed.
(□+□)÷5(\square + \square) \div 5
It survives all three, so it is the rule.

4Apply it to the last pair

#5 Look for a Pattern 6.NS.B.3
9 plus 25 is 34, divided by 5.
(9+25)÷5=6.8(9 + 25) \div 5 = 6.8
The blank is 6.8.
Answer: 6.8
4 · Reviewdoes it hold up?

Going backwards, 6.8 times 5 is 34, which is exactly 9 plus 25.

Another way: Subtracting or multiplying the pairs also gets tried, but 4 and 8 would then give 4 or 32, and neither is 2.4.

Standardsmin grade 6
  • 6.EE.A.2 Write, read, and evaluate expressions with variables — Writing the rule as an expression rather than describing it.
  • 5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Reading the pairs and results as a pattern.
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing a whole number to a decimal result.
💡Takeaway. A rule that fits one example is a guess. A rule that fits all of them is a rule.
Variant 5 medium answer: 23

Find the rule and work out the number that belongs in the empty space.

25 23 24 3 6 4.5 8 9 8.5 29 17
Show solution
1 · Understandwhat's really being asked

Three pairs of numbers each have a result written under them, and the fourth pair's result is missing. We need the rule, and then the missing number.

Givens
  • 25, 23 gives 24.
  • 3, 6 gives 4.5.
  • 8, 9 gives 8.5.
  • The pair 29, 17 has no result yet.
Unknowns
  • The number under the last pair.
Constraints
  • The same rule applies to every group.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check#13 Convert to Algebra

Guess a rule from one group and it will usually fit that group alone. Test each guess against all three finished groups before using it, and start with the sums, which are easy to compare with the results.

3 · Execute4 carry out the plan

1Add each pair

#6 Guess and Check 5.OA.B.3
Adding is the first thing to try, and it makes the three results easy to line up.
25+23=48,3+6=9,8+9=1725 + 23 = 48 ,\quad 3 + 6 = 9 ,\quad 8 + 9 = 17
Three sums to compare against three results.

2Compare each sum with its result

#5 Look for a Pattern 6.NS.B.3
48 gives 24 and 9 gives 4.5; each result is the sum divided by the same number.
48÷24=2,9÷4.5=248 \div 24 = 2,\quad 9 \div 4.5 = 2
Every group divides by 2.

3State the rule and test it

#13 Convert to Algebra 6.EE.A.2
The rule is (first + second) divided by 2. Checking it on the third group: 8 + 9 is 17, and 17 over 2 is 8.5 -- which is what is printed.
(□+□)÷2(\square + \square) \div 2
It survives all three, so it is the rule.

4Apply it to the last pair

#5 Look for a Pattern 6.NS.B.3
29 plus 17 is 46, divided by 2.
(29+17)÷2=23(29 + 17) \div 2 = 23
The blank is 23.
Answer: 23
4 · Reviewdoes it hold up?

Going backwards, 23 times 2 is 46, which is exactly 29 plus 17.

Another way: Subtracting or multiplying the pairs also gets tried, but 25 and 23 would then give 2 or 575, and neither is 24.

Standardsmin grade 6
  • 6.EE.A.2 Write, read, and evaluate expressions with variables — Writing the rule as an expression rather than describing it.
  • 5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Reading the pairs and results as a pattern.
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing a whole number to a decimal result.
💡Takeaway. A rule that fits one example is a guess. A rule that fits all of them is a rule.
Variant 6 medium answer: 8.5

Find the rule and work out the number that belongs in the empty space.

30 6 9 4 7 2.75 23 29 13 15 19
Show solution
1 · Understandwhat's really being asked

Three pairs of numbers each have a result written under them, and the fourth pair's result is missing. We need the rule, and then the missing number.

Givens
  • 30, 6 gives 9.
  • 4, 7 gives 2.75.
  • 23, 29 gives 13.
  • The pair 15, 19 has no result yet.
Unknowns
  • The number under the last pair.
Constraints
  • The same rule applies to every group.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check#13 Convert to Algebra

Guess a rule from one group and it will usually fit that group alone. Test each guess against all three finished groups before using it, and start with the sums, which are easy to compare with the results.

3 · Execute4 carry out the plan

1Add each pair

#6 Guess and Check 5.OA.B.3
Adding is the first thing to try, and it makes the three results easy to line up.
30+6=36,4+7=11,23+29=5230 + 6 = 36 ,\quad 4 + 7 = 11 ,\quad 23 + 29 = 52
Three sums to compare against three results.

2Compare each sum with its result

#5 Look for a Pattern 6.NS.B.3
36 gives 9 and 11 gives 2.75; each result is the sum divided by the same number.
36÷9=4,11÷2.75=436 \div 9 = 4,\quad 11 \div 2.75 = 4
Every group divides by 4.

3State the rule and test it

#13 Convert to Algebra 6.EE.A.2
The rule is (first + second) divided by 4. Checking it on the third group: 23 + 29 is 52, and 52 over 4 is 13 -- which is what is printed.
(□+□)÷4(\square + \square) \div 4
It survives all three, so it is the rule.

4Apply it to the last pair

#5 Look for a Pattern 6.NS.B.3
15 plus 19 is 34, divided by 4.
(15+19)÷4=8.5(15 + 19) \div 4 = 8.5
The blank is 8.5.
Answer: 8.5
4 · Reviewdoes it hold up?

Going backwards, 8.5 times 4 is 34, which is exactly 15 plus 19.

Another way: Subtracting or multiplying the pairs also gets tried, but 30 and 6 would then give 24 or 180, and neither is 9.

Standardsmin grade 6
  • 6.EE.A.2 Write, read, and evaluate expressions with variables — Writing the rule as an expression rather than describing it.
  • 5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Reading the pairs and results as a pattern.
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing a whole number to a decimal result.
💡Takeaway. A rule that fits one example is a guess. A rule that fits all of them is a rule.
Variant 7 medium answer: 1.55

Find the rule and work out the number that belongs in the empty space.

10 26 1.8 26 30 2.8 24 23 2.35 17 14
Show solution
1 · Understandwhat's really being asked

Three pairs of numbers each have a result written under them, and the fourth pair's result is missing. We need the rule, and then the missing number.

Givens
  • 10, 26 gives 1.8.
  • 26, 30 gives 2.8.
  • 24, 23 gives 2.35.
  • The pair 17, 14 has no result yet.
Unknowns
  • The number under the last pair.
Constraints
  • The same rule applies to every group.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check#13 Convert to Algebra

Guess a rule from one group and it will usually fit that group alone. Test each guess against all three finished groups before using it, and start with the sums, which are easy to compare with the results.

3 · Execute4 carry out the plan

1Add each pair

#6 Guess and Check 5.OA.B.3
Adding is the first thing to try, and it makes the three results easy to line up.
10+26=36,26+30=56,24+23=4710 + 26 = 36 ,\quad 26 + 30 = 56 ,\quad 24 + 23 = 47
Three sums to compare against three results.

2Compare each sum with its result

#5 Look for a Pattern 6.NS.B.3
36 gives 1.8 and 56 gives 2.8; each result is the sum divided by the same number.
36÷1.8=20,56÷2.8=2036 \div 1.8 = 20,\quad 56 \div 2.8 = 20
Every group divides by 20.

3State the rule and test it

#13 Convert to Algebra 6.EE.A.2
The rule is (first + second) divided by 20. Checking it on the third group: 24 + 23 is 47, and 47 over 20 is 2.35 -- which is what is printed.
(□+□)÷20(\square + \square) \div 20
It survives all three, so it is the rule.

4Apply it to the last pair

#5 Look for a Pattern 6.NS.B.3
17 plus 14 is 31, divided by 20.
(17+14)÷20=1.55(17 + 14) \div 20 = 1.55
The blank is 1.55.
Answer: 1.55
4 · Reviewdoes it hold up?

Going backwards, 1.55 times 20 is 31, which is exactly 17 plus 14.

Another way: Subtracting or multiplying the pairs also gets tried, but 10 and 26 would then give 16 or 260, and neither is 1.8.

Standardsmin grade 6
  • 6.EE.A.2 Write, read, and evaluate expressions with variables — Writing the rule as an expression rather than describing it.
  • 5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Reading the pairs and results as a pattern.
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing a whole number to a decimal result.
💡Takeaway. A rule that fits one example is a guess. A rule that fits all of them is a rule.
Variant 8 medium answer: 5

Find the rule and work out the number that belongs in the empty space.

31 20 10.2 21 10 6.2 17 3 4 6 19
Show solution
1 · Understandwhat's really being asked

Three pairs of numbers each have a result written under them, and the fourth pair's result is missing. We need the rule, and then the missing number.

Givens
  • 31, 20 gives 10.2.
  • 21, 10 gives 6.2.
  • 17, 3 gives 4.
  • The pair 6, 19 has no result yet.
Unknowns
  • The number under the last pair.
Constraints
  • The same rule applies to every group.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check#13 Convert to Algebra

Guess a rule from one group and it will usually fit that group alone. Test each guess against all three finished groups before using it, and start with the sums, which are easy to compare with the results.

3 · Execute4 carry out the plan

1Add each pair

#6 Guess and Check 5.OA.B.3
Adding is the first thing to try, and it makes the three results easy to line up.
31+20=51,21+10=31,17+3=2031 + 20 = 51 ,\quad 21 + 10 = 31 ,\quad 17 + 3 = 20
Three sums to compare against three results.

2Compare each sum with its result

#5 Look for a Pattern 6.NS.B.3
51 gives 10.2 and 31 gives 6.2; each result is the sum divided by the same number.
51÷10.2=5,31÷6.2=551 \div 10.2 = 5,\quad 31 \div 6.2 = 5
Every group divides by 5.

3State the rule and test it

#13 Convert to Algebra 6.EE.A.2
The rule is (first + second) divided by 5. Checking it on the third group: 17 + 3 is 20, and 20 over 5 is 4 -- which is what is printed.
(□+□)÷5(\square + \square) \div 5
It survives all three, so it is the rule.

4Apply it to the last pair

#5 Look for a Pattern 6.NS.B.3
6 plus 19 is 25, divided by 5.
(6+19)÷5=5(6 + 19) \div 5 = 5
The blank is 5.
Answer: 5
4 · Reviewdoes it hold up?

Going backwards, 5 times 5 is 25, which is exactly 6 plus 19.

Another way: Subtracting or multiplying the pairs also gets tried, but 31 and 20 would then give 11 or 620, and neither is 10.2.

Standardsmin grade 6
  • 6.EE.A.2 Write, read, and evaluate expressions with variables — Writing the rule as an expression rather than describing it.
  • 5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Reading the pairs and results as a pattern.
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing a whole number to a decimal result.
💡Takeaway. A rule that fits one example is a guess. A rule that fits all of them is a rule.
Variant 9 hard answer: 7

Find the rule and work out the number that belongs in the empty space.

21 32 10.6 12 32 8.8 23 2 5 12 23
Show solution
1 · Understandwhat's really being asked

Three pairs of numbers each have a result written under them, and the fourth pair's result is missing. We need the rule, and then the missing number.

Givens
  • 21, 32 gives 10.6.
  • 12, 32 gives 8.8.
  • 23, 2 gives 5.
  • The pair 12, 23 has no result yet.
Unknowns
  • The number under the last pair.
Constraints
  • The same rule applies to every group.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check#13 Convert to Algebra

Guess a rule from one group and it will usually fit that group alone. Test each guess against all three finished groups before using it, and start with the sums, which are easy to compare with the results.

3 · Execute4 carry out the plan

1Add each pair

#6 Guess and Check 5.OA.B.3
Adding is the first thing to try, and it makes the three results easy to line up.
21+32=53,12+32=44,23+2=2521 + 32 = 53 ,\quad 12 + 32 = 44 ,\quad 23 + 2 = 25
Three sums to compare against three results.

2Compare each sum with its result

#5 Look for a Pattern 6.NS.B.3
53 gives 10.6 and 44 gives 8.8; each result is the sum divided by the same number.
53÷10.6=5,44÷8.8=553 \div 10.6 = 5,\quad 44 \div 8.8 = 5
Every group divides by 5.

3State the rule and test it

#13 Convert to Algebra 6.EE.A.2
The rule is (first + second) divided by 5. Checking it on the third group: 23 + 2 is 25, and 25 over 5 is 5 -- which is what is printed.
(□+□)÷5(\square + \square) \div 5
It survives all three, so it is the rule.

4Apply it to the last pair

#5 Look for a Pattern 6.NS.B.3
12 plus 23 is 35, divided by 5.
(12+23)÷5=7(12 + 23) \div 5 = 7
The blank is 7.
Answer: 7
4 · Reviewdoes it hold up?

Going backwards, 7 times 5 is 35, which is exactly 12 plus 23.

Another way: Subtracting or multiplying the pairs also gets tried, but 21 and 32 would then give 11 or 672, and neither is 10.6.

Standardsmin grade 6
  • 6.EE.A.2 Write, read, and evaluate expressions with variables — Writing the rule as an expression rather than describing it.
  • 5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Reading the pairs and results as a pattern.
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing a whole number to a decimal result.
💡Takeaway. A rule that fits one example is a guess. A rule that fits all of them is a rule.
Variant 10 hard answer: 14.75

Find the rule and work out the number that belongs in the empty space.

6 21 6.75 28 22 12.5 15 18 8.25 27 32
Show solution
1 · Understandwhat's really being asked

Three pairs of numbers each have a result written under them, and the fourth pair's result is missing. We need the rule, and then the missing number.

Givens
  • 6, 21 gives 6.75.
  • 28, 22 gives 12.5.
  • 15, 18 gives 8.25.
  • The pair 27, 32 has no result yet.
Unknowns
  • The number under the last pair.
Constraints
  • The same rule applies to every group.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check#13 Convert to Algebra

Guess a rule from one group and it will usually fit that group alone. Test each guess against all three finished groups before using it, and start with the sums, which are easy to compare with the results.

3 · Execute4 carry out the plan

1Add each pair

#6 Guess and Check 5.OA.B.3
Adding is the first thing to try, and it makes the three results easy to line up.
6+21=27,28+22=50,15+18=336 + 21 = 27 ,\quad 28 + 22 = 50 ,\quad 15 + 18 = 33
Three sums to compare against three results.

2Compare each sum with its result

#5 Look for a Pattern 6.NS.B.3
27 gives 6.75 and 50 gives 12.5; each result is the sum divided by the same number.
27÷6.75=4,50÷12.5=427 \div 6.75 = 4,\quad 50 \div 12.5 = 4
Every group divides by 4.

3State the rule and test it

#13 Convert to Algebra 6.EE.A.2
The rule is (first + second) divided by 4. Checking it on the third group: 15 + 18 is 33, and 33 over 4 is 8.25 -- which is what is printed.
(□+□)÷4(\square + \square) \div 4
It survives all three, so it is the rule.

4Apply it to the last pair

#5 Look for a Pattern 6.NS.B.3
27 plus 32 is 59, divided by 4.
(27+32)÷4=14.75(27 + 32) \div 4 = 14.75
The blank is 14.75.
Answer: 14.75
4 · Reviewdoes it hold up?

Going backwards, 14.75 times 4 is 59, which is exactly 27 plus 32.

Another way: Subtracting or multiplying the pairs also gets tried, but 6 and 21 would then give 15 or 126, and neither is 6.75.

Standardsmin grade 6
  • 6.EE.A.2 Write, read, and evaluate expressions with variables — Writing the rule as an expression rather than describing it.
  • 5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Reading the pairs and results as a pattern.
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing a whole number to a decimal result.
💡Takeaway. A rule that fits one example is a guess. A rule that fits all of them is a rule.
Variant 11 hard answer: 7.6

Find the rule and work out the number that belongs in the empty space.

14 8 4.4 15 33 9.6 13 8 4.2 15 23
Show solution
1 · Understandwhat's really being asked

Three pairs of numbers each have a result written under them, and the fourth pair's result is missing. We need the rule, and then the missing number.

Givens
  • 14, 8 gives 4.4.
  • 15, 33 gives 9.6.
  • 13, 8 gives 4.2.
  • The pair 15, 23 has no result yet.
Unknowns
  • The number under the last pair.
Constraints
  • The same rule applies to every group.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check#13 Convert to Algebra

Guess a rule from one group and it will usually fit that group alone. Test each guess against all three finished groups before using it, and start with the sums, which are easy to compare with the results.

3 · Execute4 carry out the plan

1Add each pair

#6 Guess and Check 5.OA.B.3
Adding is the first thing to try, and it makes the three results easy to line up.
14+8=22,15+33=48,13+8=2114 + 8 = 22 ,\quad 15 + 33 = 48 ,\quad 13 + 8 = 21
Three sums to compare against three results.

2Compare each sum with its result

#5 Look for a Pattern 6.NS.B.3
22 gives 4.4 and 48 gives 9.6; each result is the sum divided by the same number.
22÷4.4=5,48÷9.6=522 \div 4.4 = 5,\quad 48 \div 9.6 = 5
Every group divides by 5.

3State the rule and test it

#13 Convert to Algebra 6.EE.A.2
The rule is (first + second) divided by 5. Checking it on the third group: 13 + 8 is 21, and 21 over 5 is 4.2 -- which is what is printed.
(□+□)÷5(\square + \square) \div 5
It survives all three, so it is the rule.

4Apply it to the last pair

#5 Look for a Pattern 6.NS.B.3
15 plus 23 is 38, divided by 5.
(15+23)÷5=7.6(15 + 23) \div 5 = 7.6
The blank is 7.6.
Answer: 7.6
4 · Reviewdoes it hold up?

Going backwards, 7.6 times 5 is 38, which is exactly 15 plus 23.

Another way: Subtracting or multiplying the pairs also gets tried, but 14 and 8 would then give 6 or 112, and neither is 4.4.

Standardsmin grade 6
  • 6.EE.A.2 Write, read, and evaluate expressions with variables — Writing the rule as an expression rather than describing it.
  • 5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Reading the pairs and results as a pattern.
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing a whole number to a decimal result.
💡Takeaway. A rule that fits one example is a guess. A rule that fits all of them is a rule.
Variant 12 hard answer: 1.8

Find the rule and work out the number that belongs in the empty space.

18 8 2.6 12 20 3.2 23 33 5.6 4 14
Show solution
1 · Understandwhat's really being asked

Three pairs of numbers each have a result written under them, and the fourth pair's result is missing. We need the rule, and then the missing number.

Givens
  • 18, 8 gives 2.6.
  • 12, 20 gives 3.2.
  • 23, 33 gives 5.6.
  • The pair 4, 14 has no result yet.
Unknowns
  • The number under the last pair.
Constraints
  • The same rule applies to every group.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check#13 Convert to Algebra

Guess a rule from one group and it will usually fit that group alone. Test each guess against all three finished groups before using it, and start with the sums, which are easy to compare with the results.

3 · Execute4 carry out the plan

1Add each pair

#6 Guess and Check 5.OA.B.3
Adding is the first thing to try, and it makes the three results easy to line up.
18+8=26,12+20=32,23+33=5618 + 8 = 26 ,\quad 12 + 20 = 32 ,\quad 23 + 33 = 56
Three sums to compare against three results.

2Compare each sum with its result

#5 Look for a Pattern 6.NS.B.3
26 gives 2.6 and 32 gives 3.2; each result is the sum divided by the same number.
26÷2.6=10,32÷3.2=1026 \div 2.6 = 10,\quad 32 \div 3.2 = 10
Every group divides by 10.

3State the rule and test it

#13 Convert to Algebra 6.EE.A.2
The rule is (first + second) divided by 10. Checking it on the third group: 23 + 33 is 56, and 56 over 10 is 5.6 -- which is what is printed.
(□+□)÷10(\square + \square) \div 10
It survives all three, so it is the rule.

4Apply it to the last pair

#5 Look for a Pattern 6.NS.B.3
4 plus 14 is 18, divided by 10.
(4+14)÷10=1.8(4 + 14) \div 10 = 1.8
The blank is 1.8.
Answer: 1.8
4 · Reviewdoes it hold up?

Going backwards, 1.8 times 10 is 18, which is exactly 4 plus 14.

Another way: Subtracting or multiplying the pairs also gets tried, but 18 and 8 would then give 10 or 144, and neither is 2.6.

Standardsmin grade 6
  • 6.EE.A.2 Write, read, and evaluate expressions with variables — Writing the rule as an expression rather than describing it.
  • 5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Reading the pairs and results as a pattern.
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing a whole number to a decimal result.
💡Takeaway. A rule that fits one example is a guess. A rule that fits all of them is a rule.