Number Patterns and Missing Numbers

Skip counting means counting with jumps of the same size. You can jump forward or back. A number line helps you see where each jump lands.

Choose your start and jump

Start at 0 and count by 5s: 0, 5, 10, 15, 20, 25. Each number is 5 more than the last. Start at 20 and count back by 2s: 20, 18, 16, 14, 12. Each number is 2 less than the last.

You can start anywhere. Counting by 10s from 7 gives 7, 17, 27, 37, 47. The ones digit stays 7. Counting by 3s from 1 gives 1, 4, 7, 10, 13. These are different from the count that starts at 0.

Try different jump sizes

Read each row across. The gap stays the same within that row.

Numbers reached by counting from 0 in equal jumps are multiples of the jump size. You meet these numbers in times tables too.

For bigger starts, change the right place value. Adding 100 to 235 gives 335. Keep going: 435, 535, 635. The tens and ones stay the same. Counting back by 100s from 862 gives 862, 762, 662, 562.

Cross a hundred carefully

Count by 10s: 170, 180, 190, 200, 210. After 190, one more ten makes 200. Counting back gives 420, 410, 400, 390. Taking one ten from 400 leaves 390.

With 5s, you might pass through 185, 190, 195, 200, 205. Keep the jump the same as you cross 200. Counting back by 50s from 1,000 gives 1,000, 950, 900, 850, 800.

Show the count

On a number line, mark the starting number. Draw one equal hop for each jump. Label where you land, including the first landing.

On a hundred chart, each row has 10 numbers. Moving down one square adds 10. Counting by ones moves right across a row. At the row's end, continue at the first square of the next row.

A number pattern, or sequence, is an ordered list that follows a rule. Each number is a term. In 4, 7, 10, 13, the first term is 4. The fourth term is 13.

Look between neighbors

A term-to-term rule tells you how to get the next term. Look at numbers next to each other. Ask whether they go up or down. Then work out the change.

For 4, 7, 10, 13:

1. From 4 to 7, add 3.
2. From 7 to 10, add 3.
3. From 10 to 13, add 3.

The rule is "add 3." Use it on the last term: $13+3=16$. Use it again: $16+3=19$. The next two terms are 16 and 19.

In 85, 80, 75, 70, each term is 5 less. The rule is "subtract 5." The next two terms are 65 and 60. A pattern that gets bigger is growing. One that gets smaller is shrinking.

Check more than one jump

Look at 2, 4, 8, 16. The first jump is 2. But adding 2 to 4 gives 6, so "add 2" fails. Here the jumps are 2, 4, and 8. Each term is double the last:

$2\times 2=4$, then $4\times 2=8$, then $8\times 2=16$.

The rule is "double," or "multiply by 2." The next term is 32.

Two terms can fit different rules. The start 3, 6 fits both "add 3" and "double." A third term helps you test them. If it is 9, "add 3" fits and "double" does not.

Name the start and the rule

"Start at 12 and add 4 each time" gives 12, 16, 20, 24, 28. The start tells you where to begin. The rule tells you what to do next.

Saying "the rule is 12" leaves out the change. It also leaves someone unsure what to write after 12. Say the action and its amount: "add 4."

To find several later terms, keep using the rule on your newest term. For 46, 56, 66, 76, add 10 three more times. You get 86, 96, 106, crossing 100 with the same jump.

A box can hide a number anywhere in a pattern. Find the rule from the numbers you can see. Then use it to fill the gap.

Check both sides of a box

Look at 14, 20, □, 32, 38. The jump from 14 to 20 is 6. The jump from 32 to 38 is also 6. With the rule "add 6," the missing number is $20+6=26$. Check the next jump: $26+6=32$. Both sides fit.

For a missing first term, work backward. In □, 45, 55, 65, the forward rule is "add 10." Take 10 from 45 to find 35. In □, 72, 70, 68, the forward rule is "subtract 2." Add 2 to 72 to find 74.

Count the jumps across a gap

In an adding pattern, one missing term makes two jumps. Try 5, □, 13 with equal jumps.

1. Find the whole gap: $13-5=8$.
2. Split 8 into two equal jumps: $8÷2=4$.
3. Jump once from 5: $5+4=9$.

The pattern is 5, 9, 13. Answering 8 mixes up the whole gap with the missing number. The missing number must sit between 5 and 13.

Two boxes in a row make three jumps. In 10, □, □, 25, the whole gap is 15. Three jumps of 5 make 15. Land on 15, then 20, then 25.

Equal gaps work for adding or subtracting the same amount. A doubling pattern works differently. In 2, □, 8 with the rule "double," the box is 4. The jumps are 2 and 4, so they are different sizes.

Fill a shrinking pattern

For 90, 75, □, 45, □, subtract 15 each time. From 75, subtract 15 to get 60. From 45, subtract 15 to get 30. Read it through: 90, 75, 60, 45, 30.

Find the number that breaks the rule

The pattern should add 5: 30, 35, 40, 44, 50, 55. The jumps around 44 are wrong. From 40 to 44 is 4. From 44 to 50 is 6.

Replace 44 with 45. Now $40+5=45$ and $45+5=50$. Read from the start to check every jump, including the ones beside your repair.

You can make a pattern by choosing its start and rule. Pictures can show number patterns too. Count their pieces to see what changes.

Follow a rule like a recipe

Start at 9 and add 5 each time: 9, 14, 19, 24, 29, 34. Write 9 first. Then add 5 to the last number you wrote. The starting number already counts as the first term.

Changing the rule changes the pattern. Start at 2 and add 2: 2, 4, 6, 8, 10. Start at 2 and add 3: 2, 5, 8, 11, 14. With the same start, the different jumps lead to different terms.

A multiplying rule works too. Start at 5 and double: 5, 10, 20, 40, 80. Always double the last number, including the new one you just found.

Turn pictures into numbers

Imagine an L made from dots. The first L has 3 dots. Add one dot at the end of each arm. The next L has 5 dots. Do that again to get 7 dots. Each new picture has 2 extra dots.

The picture number tells you which drawing you mean. The dots column tells you how many dots it has. To draw picture 4, start with picture 3. Add one dot to each arm, making 9 altogether. You add 2 dots because two arms each get one.

Find the group that repeats

Some patterns return to the same numbers. In 1, 4, 2, 1, 4, 2, 1, 4, 2, the group 1, 4, 2 keeps coming back. This group is the unit of repeat. Keep its numbers in the same order.

To find the 10th term of the repeating pattern, count whole groups. Terms 1–3 are 1, 4, 2. Terms 4–6 and 7–9 repeat that group. Term 10 starts the next group, so it is 1. Jumping straight to 10 without tracking the groups can put you in the wrong place.

Put 8 counters into pairs. You get 4 pairs with none left. Eight is even. Pair up 7 counters and one is left without a partner. Seven is odd.

Spot odd and even quickly

You can use the ones digit, even for a large number.

The number 358 ends in 8, so it is even. The number 461 ends in 1, so it is odd. You do not need to draw hundreds of counters.

The last digit works because full tens can all be paired. In 30, there are three full tens and no extra ones. Each ten splits into 5 pairs. Nothing is left over, so 30 is even.

Look at the last digit

The first digit can trick you. The number 52 starts with 5, but ends in 2. It is even. The number 29 starts with 2, but ends in 9. It is odd. Adding hundreds does not change this: 74 and 574 are both even.

Zero is even too. Zero things leave nothing unpaired. Zero also starts the even count 0, 2, 4, 6, 8.

Watch neighbors take turns

Count by ones: 25, 26, 27, 28. You go odd, even, odd, even. Adding one changes whether something is left over. After an even number, the next counter has no partner. After an odd number, one more counter pairs with the leftover.

The numbers just before and after 28 are 27 and 29. Both are odd. The neighbors of an odd number are even. For example, 40 and 42 sit beside 41.

Connect even numbers to doubles

Every even number is double a whole number. Fourteen is double 7: $7+7=14$. You can split 14 things into two equal groups of 7.

An odd number is one more than an even number. Fifteen is double 7 with one extra: $14+1=15$. Two groups of 7 use 14 things, leaving one without a match.

You can predict whether an answer is odd or even. Think about pairs and leftovers before you calculate.

Join pairs and leftovers

For $6+8=14$, pairs join more pairs. For $5+7=12$, the leftover from 5 pairs with the leftover from 7. For $9+4=13$, the leftover from 9 stays alone.

Subtraction has matching patterns. Even minus even gives even, as in $20-6=14$. Odd minus odd gives even, as in $15-9=6$. Odd minus even and even minus odd give odd answers: $17-4=13$ and $12-5=7$.

Use the prediction as a check

Suppose you write $37+25=61$. Both starting numbers are odd. Their sum must be even, so 61 cannot be right. Add carefully to get 62.

Getting the right kind of number is only one check. The answer must also be the correct amount. The number 64 is even too, but it is not the sum of 37 and 25.

Explain a whole pattern

Adding an even number keeps odd or even unchanged. Start at 7 and add 2: 7, 9, 11, 13, 15. Every term is odd. Adding a pair leaves the single leftover in place.

Adding an odd number swaps odd and even each time. Start at 1 and add 3: 1, 4, 7, 10, 13, 16. An odd start plus an odd jump makes even. Then an even term plus an odd jump makes odd.

Counting by 5s from 0 gives 0, 5, 10, 15, 20. These go even, odd, even, odd, even, because 5 is odd.

See why four groups make an even total

Four equal groups can split into two equal halves. With 4 groups of 7, put 2 groups in each half. Each half has 14, so $4\times 7=28$ and $14+14=28$. Four times any whole number is even because the groups always split this way.

Charts put numbers in useful places. You can explain a pattern by thinking about what changes as you move.

Move on a hundred chart

A hundred chart has 10 numbers in each row. The first row runs from 1 to 10. The next runs from 11 to 20.

Within a row, one square right adds 1. One square down adds 10. A column such as 3, 13, 23, 33, 43 keeps the same ones digit. Adding 10 changes the tens while leaving the ones alone.

A diagonal step down and right adds 11: 12, 23, 34, 45. It combines 10 down with 1 right. A diagonal step down and left adds 9: 18, 27, 36, 45. It combines 10 down with 1 back. These moves stay inside the chart's edges.

Notice times table endings

A 10 is two 5s and five 2s. So a multiple of 10 is also a multiple of 5 and 2. For example, 30 is 3 tens, 6 fives, or 15 twos.

Compare the same number of groups. Six groups of 10 make 60. Six groups of 5 make 30, half as much. Each group is half the size. Four groups of a number make double what two groups make: $2\times 6=12$, so $4\times 6=24$.

Read an addition table

In an addition table, add the row label to the column label. Here is a small part:

The row for 2 and column for 3 meet at 5. Moving one square right increases one addend by 1. The answer also increases by 1.

Across the diagonal, matching squares have the same answers. The row for 2, column for 3 gives the same as row 3, column 2. That is because $2+3=3+2$. Along the diagonal itself, the labels match. These are doubles, so every answer there is even.

The equal sign means "is the same amount as." Read $3+4=7$ as "3 plus 4 is the same amount as 7."

Compare both sides

Think of a balance scale with two pans. Put the value of each side in its own pan. Equal amounts make the pans level. For $6+2=8$, each pan holds 8.

A calculation can be on either side. The sentence $9=4+5$ is true because each side is 9. So is $10=10$. In $2+7=5+4$, both sides have calculations, and both make 9.

To decide whether a sentence is true, work out each side separately. For $6=10-4$, compare 6 with 6. It is true. For $5+1=6+2$, compare 6 with 8. It is false.

Keep each equal sign true

An equal sign does not mean "write the next answer here." In $8+4=\mathrm{\square }+5$, writing 12 in the box gives $8+4=12+5$. The left is 12, but the right is 17. They do not balance.

Be careful when showing several steps. This chain is correct:

Every part is worth 28. But $20+5=25+3=28$ is false: the first part is 25 and the next is 28. You can write two separate sentences instead: $20+5=25$, then $25+3=28$.

Switch order where it works

Adding the same numbers in either order gives the same total: $3+5=5+3$. Multiplication can switch order too: $2\times 5=5\times 2$. Also, $4+4=2\times 4$, because both show two groups of 4.

Subtraction and division depend on order. Taking 2 from 10 leaves 8. You cannot swap them and still get 8. Likewise, $10÷2=5$, but dividing 2 into 10 equal groups does not give 5.

Compare unequal sides

The sign $<$ means "is less than." The sign $>$ means "is greater than." The sign $\mathrm{\ne }$ means "is not equal to."

$9+6>14$ is true because 15 is greater than 14. $12-5<8$ is true because 7 is less than 8. $4+4\mathrm{\ne }9$ is true because 8 and 9 differ.

When a number sentence has a box, the missing number must make it true. Work out a complete side first. Then make the other side match.

A box is only part of its side

Solve $8+4=\mathrm{\square }+5$.

1. Work out the left side: $8+4=12$.
2. The right side must also make 12.
3. Find what joins 5 to make 12: $12-5=7$.
4. Check: $7+5=12$. The box is 7.

Writing 12 in the box gives 17 on the right. Writing 17 adds all three numbers together.

Notice how numbers are linked

In $18+\mathrm{\square }=17+6$, the 18 is 1 more than 17. To keep the totals equal, the box must be 1 less than 6. It is 5. Check both totals: $18+5=23$ and $17+6=23$.

You can move 1 from one addend to the other. For $27+9=26+10$, one number gets 1 smaller and the other gets 1 bigger. The sum stays 36. The right side is easier to add mentally.

Subtraction works differently. Change both numbers by the same amount to keep the gap the same. In $43-19=44-20$, both numbers go up by 1. Both differences are 24. You can picture two points on a number line sliding one place together; their distance stays the same.

Balance different operations

For $2\times 6=\mathrm{\square }+2$, the left side is 12. The box must be 10, because $10+2=12$.

For $5\times 4=30-\mathrm{\square }$, the left side is 20. Take 10 from 30 to make 20, so the box is 10.

Change a whole sentence fairly

Start with $3+4=7$. Add 2 to each side: $3+4+2=7+2$. Each side is now 9. Adding to only one side would tip the balance. Taking the same amount from both sides also keeps them equal.

You can compare sides that do not match too. For $25+9$ and $30+6$, the values are 34 and 36. Since 34 is smaller, write $25+9<30+6$.

Inverse operations undo each other. Add 15 to 23 and you get 38. Take the same 15 away from 38 and you are back at 23:

$23+15=38$ and $38-15=23$.

Make a fact family

The numbers 6, 9, and 15 fit together. Six and nine are the parts. Fifteen is the whole. Their fact family has two addition facts and two subtraction facts:

• $6+9=15$
• $9+6=15$
• $15-9=6$
• $15-6=9$

In either addition, joining the parts makes the whole. In either subtraction, start with the whole and take one part away. The other part is left.

Picture the parts

Draw one long bar labeled 15. Split it into a part labeled 6 and a part labeled 9. The full length shows the whole. Cover the 6 part and the 9 part remains. Cover the 9 part and the 6 part remains.

This is a bar model. It helps you decide which numbers to join and which to take away. If you know the whole and one part, you can find the hidden part.

The order matters in subtraction. The whole comes first. The sentence $15-6=9$ fits the model. Swapping 15 and 6 does not leave 9: you cannot take a bar of 15 from a bar of 6 and have 9 left.

Check with the inverse

Suppose you calculate $52-17=35$. To check, add the part you took away back to the part left:

You reach the starting whole, so the subtraction checks. Repeating the same subtraction can repeat the same slip. Addition gives you a different way to check the parts.

For an addition, try taking one part from your answer. If $23+15=38$, then $38-23=15$. The part you get back should match.

Fact families work with hundreds too. The addition $240+130=370$ gives $370-130=240$ and $370-240=130$. The whole is 370, even when it appears after the equal sign.

A box, question mark, line, or shape can hide a number. Look at what the missing number does in the sentence. Is it a part or the whole?

Find a part in addition

For $8+\mathrm{\square }=11$, the whole is 11. One part is 8. Take that part from the whole: $11-8=3$. Check by putting 3 in the box: $8+3=11$.

You can also count on from 8 to 11. Land on 9, then 10, then 11. That takes 3 jumps of one.

The box can be first. In $\mathrm{\square }+26=50$, take 26 from 50: $50-26=24$. Check: $24+26=50$. Its place on the page does not make the box the whole.

Find what was taken away

In $15-\mathrm{\square }=9$, the starting whole is 15. Nine is what remains. The box is the part taken away. Find it with $15-9=6$. Check: $15-6=9$.

Find the starting whole

In $\mathrm{\square }-6=7$, you know both parts. Six was taken away and 7 remains. Join them: $7+6=13$. Check: $13-6=7$.

Subtracting the two visible numbers would give 1. But the start must be more than 7: after taking 6 away, 7 is still left. Picture putting the missing 6 back beside the 7.

Read either way around

The sentence $8=\mathrm{\square }-5$ says the same as $\mathrm{\square }-5=8$. Add 5 to 8 to find the starting whole, 13. In $14=9+\mathrm{\square }$, the whole is 14. Subtract 9 to find 5.

Use the same method with bigger numbers

For $56+\mathrm{\square }=182$, take 56 from 182. Take 50 to get 132, then 6 to get 126. Check: $56+126=182$.

For $245-\mathrm{\square }=120$, take the amount left from the start: $245-120=125$. Put it back: $245-125=120$. Here the missing number is the part removed, not the amount left.

Multiplication and division undo each other. If 5 groups of 7 make 35, splitting 35 into 5 equal groups gives 7 in each.

$5\times 7=35$ and $35÷5=7$.

Connect a fact family

With 2, 9, and 18, you can write:

• $2\times 9=18$
• $9\times 2=18$
• $18÷2=9$
• $18÷9=2$

Multiplication can switch order. For a division fact, the total comes first. Here the total is 18. Dividing 2 by 18 is a different calculation.

Find a missing group number

In $\mathrm{\square }\times 5=35$, ask how many 5s make 35. Count 5, 10, 15, 20, 25, 30, 35. Seven jumps land on 35, so the box is 7. Division gives the same answer: $35÷5=7$.

For $10\times \mathrm{\square }=80$, divide the total by 10: $80÷10=8$. Check: $10\times 8=80$. Multiplying a whole number by 10 shifts its digits one place left and puts 0 in the ones place.

Find a missing divisor

Try $30÷\mathrm{\square }=6$. The total is 30, and the answer is 6. Think of the matching multiplication: $6\times \mathrm{\square }=30$. Six groups of 5 make 30, so the box is 5. Check: $30÷5=6$.

Find the total being shared

For $\mathrm{\square }÷2=9$, there are two equal groups of 9. Join those groups: $9\times 2=18$. The box is 18. Check: $18÷2=9$.

Dividing 9 by 2 would split one of the groups. Here you need to find the full amount before sharing. Drawing two circles with 9 counters in each shows why the start is 18.

Use groups to balance both sides

Four bags hold 20 apples altogether. Each bag has the same amount. Write $4\times \mathrm{\square }=20$. Each bag holds 5 apples because $4\times 5=20$.

In $5\times \mathrm{\square }=10\times 3$, first work out the right side: $10\times 3=30$. Now find how many 5s make 30. The box is 6, and $5\times 6=30$ checks.

Sometimes a missing box needs an operation sign. Sometimes a shape hides a number. Work out which kind of puzzle you have before you start.

Try a sign and check

In $10\mathrm{\square }2=12$, test the signs:

Only addition gives 12. The missing sign is $+$.

With positive whole numbers bigger than 1, adding or multiplying makes the first number bigger. Subtracting or dividing makes it smaller. This can help you choose what to try first. You still need to check the calculation.

For $20\mathrm{\square }2=10$, try subtraction and division. Taking 2 away gives 18. Dividing by 2 gives 10, so $÷$ fits.

More than one sign can work. In $2\mathrm{\square }2=4$, addition and multiplication both fit. Check all the signs when a puzzle asks for every answer.

Match calculations on two sides

In $8\mathrm{\square }2=5\mathrm{\square }2$, each box needs a sign. The left can make 10, 6, 16, or 4. On the right, addition makes 7, subtraction makes 3, and multiplication makes 10. Division on the right does not give a whole number. The matching value is 10:

Keep each shape's value the same

In one puzzle, the same shape always means the same number. For $\mathrm{△}+\mathrm{△}=16$, two equal numbers make 16. Half of 16 is 8, so each triangle is 8. Check: $8+8=16$.

For $\bullet +\bullet +\bullet =15$, three equal numbers make 15. Each circle is 5 because $5+5+5=15$. You cannot give the three circles different values.

Use one clue in the next clue

A star is 6, and $\star +\mathrm{■}=10$. Replace the star with 6. Now $6+\mathrm{■}=10$, so the square is 4.

If $\diamond -4=6$, the diamond is the starting whole. Add 4 back to get 10. A later clue asking for twice the diamond gives $10\times 2=20$. Keep the diamond worth 10 through the whole puzzle.

Read a story and picture what happens. Find what you know and what you need. Put a box where the unknown number belongs.

Find the amount at the start

"Maya had some stickers. She got 8 more. Now she has 23." Her starting amount is hidden, so write $\mathrm{\square }+8=23$. Undo the extra 8: $23-8=15$. Maya started with 15 stickers. Check the story: 15 stickers and 8 more make 23.

Find a change

"Leo has 14 marbles. How many more does he need to have 30?" Write $14+\mathrm{\square }=30$. Find the missing part: $30-14=16$. Leo needs 16 more marbles.

The word "more" does not tell you to add 14 and 30. Thirty is the target total, not an extra group. Adding them would give 44, which passes the target.

"A baker made 40 muffins and sold some. There are 13 left." Write $40-\mathrm{\square }=13$. Subtract the amount left from the start: $40-13=27$. The baker sold 27 muffins.

Show equal groups

"Pencils come in packs of 5. How many packs make 40 pencils?" Write $\mathrm{\square }\times 5=40$. Divide the total by the pack size: $40÷5=8$. You need 8 packs. Check that 8 groups of 5 make 40 pencils.

Use two steps when you need them

"A class has 28 children. Twelve go to the library and 9 go to art. How many stay?" Show all three groups: $12+9+\mathrm{\square }=28$.

1. Find how many leave: $12+9=21$.
2. Find how many stay: $28-21=7$.

Seven children stay in the classroom. Adding 7 back to the 21 who left gives all 28 children.

Money can work the same way. "Ava has $20. She spends $6 on a book and $3 on a pen." The sentence is $6+3+\mathrm{\square }=20$. First find the $9 spent. Then $20-9=11$, so Ava has $11 left.

Answer and check the story

Say what the number counts: 8 packs, 27 muffins, or 7 children. Then put the answer back into the story. For the muffins, $27+13=40$ checks the total. You can also estimate: selling about 30 from 40 would leave about 10, so 27 sold with 13 left makes sense.

A pattern can help you predict a later amount. You need its starting amount and what happens each time. Keep track of both the amounts and the number of steps.

Count changes from the start

Sam has $6 and saves $4 each week. The rule is "add 4." The amounts are 6, 10, 14, 18, 22, 26, 30.

The first row is week 0, before Sam adds any money. After one week he has $10. After six weeks he reaches $30. Counting the starting row as one week would give the wrong time.

Shrinking patterns predict amounts too. Jo has 50 stickers and uses 5 each day. The amounts left are 50, 45, 40, 35, 30. Four days make four changes, so 30 stickers remain after day 4.

Find a later term

For a sequence, the first number is term 1. To find the 10th term of 3, 6, 9, 12, continue and count:

3, 6, 9, 12, 15, 18, 21, 24, 27, 30.

The 10th term is 30. There are ten terms but only nine jumps from the first to the tenth. A week-0 starting amount and a term-1 starting number use different labels. Read the labels before counting steps.

Check which numbers fit

In 2, 7, 12, 17, 22, the rule is "add 5." The ones digits take turns between 2 and 7. A number ending in 5 cannot belong to this pattern. The number 47 does belong: continue through 27, 32, 37, 42, 47.

A matching last digit is a useful clue. You can check a nearby number by continuing the pattern until you reach it or pass it.

Explain a prediction about odd and even

The claim "Every term in 5, 9, 13, 17 is odd" fits a pattern that keeps adding 4. The start is odd, and adding an even number keeps it odd.

The pattern 10, 13, 16, 19 behaves differently. It starts even and adds an odd number, 3. Its terms take turns: even, odd, even, odd. The starting amount and the jump together tell you what happens.