Unit 6 · Multiplication and Division

Maya packs muffins for a bake sale. She has 4 boxes, and each box holds 6 muffins. Every box has the same number. These are equal groups. She could count the muffins one by one. That takes a long time. Instead she counts by 6s: 6, 12, 18, 24. Four boxes hold 24 muffins.

Math has a short way to write this. We say 4 groups of 6 and write 4 × 6 = 24. The sign × means multiply. The numbers you multiply, 4 and 6, are each called a factor. The answer, 24, is the product. The first factor tells how many groups. The second factor tells how many are in each group.

Groups only count as equal if every group has the same amount. Three bags with 5, 5 and 5 apples are equal groups: 3 × 5 = 15. Three bags with 5, 4 and 6 apples are not, even though the total is still 15. A common mistake is to multiply the number of groups by the total. Always multiply the groups by the amount in one group.

Jamal walks his neighbor’s dog every day for a week. He earns $5 each time. To find his pay for 7 days, he could add. 5 + 5 + 5 + 5 + 5 + 5 + 5 = 35. That is seven 5s. Adding the same number over and over is called repeated addition. Multiplication is a faster way to write it: 7 × 5 = 35.

You can turn any repeated addition into multiplication. Count how many times the number appears. That count is the first factor. The number being added is the second factor. So 9 + 9 + 9 + 9 = 4 × 9 = 36. And 2 + 2 + 2 + 2 + 2 + 2 = 6 × 2 = 12. Check: six 2s make 12. We also say 6 times 2.

Be careful: 6 × 2 means six 2s, not 6 + 2. A student who writes 6 × 2 = 8 has added the factors instead of multiplying. Another student might write 3 × 4 as 3 + 3 + 3 + 3 and get 12. That total is still correct, because three 4s and four 3s add to the same number. But the meaning of 3 × 4 is three groups of 4.

An egg carton holds eggs in 2 rows with 6 in each row. Objects lined up in neat rows and columns make an array. A row goes across. A column goes up and down. To count the eggs, multiply the number of rows by how many are in each row: 2 × 6 = 12. A dozen eggs is 12.

A muffin tin might have 3 rows with 4 cups in each row. That array shows 3 × 4 = 12. Count the rows first, then count across one row. Do not count every cup; the array does that work for you. If you skip count by rows, you get 4, 8, 12. Same answer, and much faster than counting one by one.

Arrays are everywhere: windows on a building, seats in a theater, tiles on a floor. A classroom has 5 rows of desks with 6 desks in each row. That is 5 × 6 = 30 desks. A common mistake is adding rows and columns: 5 + 6 = 11. Eleven desks would not even fill two rows. Multiply, and 30 makes sense.

Draw an array with 3 rows of 5 dots. It shows 3 × 5 = 15. Now turn your paper a quarter turn. The same dots now sit in 5 rows of 3. That is 5 × 3 = 15. Nothing was added or taken away, so the total must be the same. Turning the array is a picture of an important rule.

The rule is the commutative property: you can multiply two numbers in either order and get the same product. 4 × 7 = 28 and 7 × 4 = 28. 2 × 9 = 18 and 9 × 2 = 18. Commutative comes from the word commute, which means to move or switch places. The factors switch places, and the product stays the same.

This rule cuts your work in half. If you know 8 × 3 = 24, you also know 3 × 8 = 24. Learn 6 × 9 = 54, and 9 × 6 comes free. But be careful: subtraction does not work this way. 9 − 2 is 7, but 2 − 9 is not 7. Switching the numbers works for addition and multiplication only.

Suppose you forget 7 × 6. Draw an array with 7 rows of 6. Now draw a line after the fifth row. The top part is 5 rows of 6, which is 30. The bottom part is 2 rows of 6, which is 12. Add the parts: 30 + 12 = 42. So 7 × 6 = 42. You split one hard fact into two easy facts.

This is the distributive property. When you break apart one factor into two parts, you multiply each part and then add. 7 × 6 = (5 × 6) + (2 × 6) = 30 + 12 = 42. The other factor, 6, gets shared out to both parts. Try 8 × 4. Break 8 into 5 and 3: (5 × 4) + (3 × 4) = 20 + 12 = 32.

Choose parts you know well. Fives, twos and tens make good parts. For 9 × 7, break 9 into 5 and 4: (5 × 7) + (4 × 7) = 35 + 28 = 63. One mistake is forgetting to multiply the second part: 35 + 4 = 39 is wrong. Every part must be multiplied by the other factor before you add.

Cover a rectangle with same-size squares. The number of squares is its area. A rug is 3 squares wide and 8 squares long. Count the squares: 3 rows of 8 is 24 squares. You do not have to count every square. Multiply the two sides: 3 × 8 = 24. The rectangle is an array made of squares.

This rectangle is called an area model. It shows a product as a shape. A 6-by-7 rectangle has an area of 42 square units. A 10-by-10 rectangle has 100 square units. The two sides are the factors, and the area is the product. If each square is 1 foot on a side, the area is measured in square feet.

You can split an area model just like an array. A 6-by-7 rectangle can be cut into a 6-by-5 piece and a 6-by-2 piece. Their areas are 30 and 12, and 30 + 12 = 42. Later you will use this picture for bigger numbers, like 6 × 27. A common mistake is adding the sides: 6 + 7 = 13 counts edges, not squares.

Multiplying by 2 means doubling: 2 × 7 is 7 + 7 = 14. Doubles are easy for most people. Here is the useful part. Four is double two. So to multiply by 4, double, then double again. For 4 × 7: double 7 is 14, and double 14 is 28. So 4 × 7 = 28. Check by skip counting: 7, 14, 21, 28.

Eight is double four. To multiply by 8, double three times. For 8 × 6: double 6 is 12, double 12 is 24, double 24 is 48. So 8 × 6 = 48. Try 8 × 9: double 9 is 18, then 36, then 72. So 8 × 9 = 72. Doubling turns the hard eights into three easy steps.

Keep track of how many times you doubled. One double is × 2, two doubles is × 4, three doubles is × 8. A student who doubles 7 twice and calls it 8 × 7 = 28 has stopped too soon. 8 × 7 needs three doubles: 14, 28, 56. Write the steps down until they feel automatic.

Count by tens: 10, 20, 30, 40. Every product of 10 ends in a zero, and the other factor sits in front. A tens fact like 10 × 7 = 70 or 10 × 3 = 30 is quick to see. Look at a dime, worth 10 cents. Six dimes are 6 × 10 = 60 cents. Tens facts are the fastest facts on the table.

Five is half of ten. So a fives fact is half of a tens fact. 5 × 8 is half of 10 × 8. Half of 80 is 40, so 5 × 8 = 40. For 5 × 7, half of 70 is 35. Notice the pattern: products of 5 end in 5 or 0. Odd numbers times 5 end in 5, like 5 × 3 = 15. Even numbers times 5 end in 0, like 5 × 4 = 20.

Nickels show fives in action. A nickel is 5 cents. Nine nickels are 9 × 5 = 45 cents. A clock shows fives too. Each number on a clock face marks 5 minutes. The minute hand at the 4 means 4 × 5 = 20 minutes past. A student who writes 5 × 6 = 35 has counted one five too many; 5 × 6 = 30, and 30 ends in 0 because 6 is even.

Nines look hard, but they hide a pattern. Write the nines facts: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90. Look at each digit. In 18, 1 + 8 = 9. In 27, 2 + 7 = 9. In 63, 6 + 3 = 9. The digits of every nines product up to 9 × 10 add to 9. That gives you a way to check your answer.

Here is the nines pattern as a shortcut. For 9 × 7, the tens digit is one less than 7, so it is 6. The ones digit must make the digits add to 9, so it is 3. The answer is 63. For 9 × 4: tens digit 3, ones digit 6, so 36. Check by adding: 3 + 6 = 9. It works for 9 × 1 through 9 × 10.

Another way uses tens. Nine is one less than ten, so 9 × 7 is 10 × 7 minus one 7. That is 70 − 7 = 63. And 9 × 8 is 80 − 8 = 72. Both routes give the same answer, which is a good check. A student who writes 9 × 6 = 56 should notice that 5 + 6 = 11, not 9. The real answer is 54.

A multiple of 3 is 3 times a whole number. Multiply 3 by 1, 2, 3 and so on: 3, 6, 9, 12, 15, 18. Skip counting lists the multiples. Every multiple of 2 is even. Every multiple of 5 ends in 5 or 0. Every multiple of 10 ends in 0. Is 24 a multiple of 4? Yes, because 4 × 6 = 24.

A multiplication table lists products in a grid. The row for 4 reads 4, 8, 12, 16, 20 and so on. Find 6 × 7 by going to the row for 6 and the column for 7. They meet at 42. Because of the commutative property, the table is a mirror across its diagonal. The row for 6 matches the column for 6.

The diagonal holds a special list: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. Each is a square number, a number times itself, like 7 × 7 = 49. Each is the area of a square. Look for more patterns. In the row for 6, every product is even. Going down any column, each number grows by the same amount.

A word problem tells a story with numbers hidden in it. Read: "A Chicago bus has 6 rows of seats on one side, with 2 seats in each row. How many seats are on that side?" Ask two questions. How many groups? 6 rows. How many in each group? 2 seats. Write 6 × 2 = 12. That side has 12 seats.

Some problems use scaling: one thing is so many times as many as another. "Ana’s ribbon is 4 feet long. Ben’s ribbon is 3 times as long." Ben’s ribbon is 3 × 4 = 12 feet. Scaling does not mean groups of ribbons. It means stretching one length to 3 times its size. Draw a bar for Ana and a bar 3 times longer for Ben.

Always check your answer against the story. "Each of 7 students brings 3 pencils. How many pencils?" If you get 10, ask: could 7 students with 3 pencils each have only 10? No. 7 × 3 = 21 pencils. Words like each, per, every and times as many are clues to multiply. Not every problem multiplies, so picture the story first.

Four friends bake 28 cookies and want to share them fairly. Fair means each friend gets the same number. Deal them out like cards: one for you, one for you, one for you, one for me. Keep going until the cookies are gone. Each friend ends up with 7. That is division as sharing, and we write 28 ÷ 4 = 7.

The sign ÷ means divide. Read 28 ÷ 4 as 28 divided by 4. The 28 is the total. The 4 is how many groups. The 7 is how many in each group. Dealing out works, but it is slow. A faster way: ask what number times 4 makes 28. Since 4 × 7 = 28, the answer is 7.

Try 30 ÷ 5. Thirty stickers are shared by 5 students. Ask: 5 times what makes 30? 5 × 6 = 30, so each student gets 6. A common mistake is to subtract: 30 − 5 = 25. Twenty-five stickers each would need 125 stickers in all. Check by multiplying back: 5 × 6 = 30, the total you started with.

Now the cookies go into bags, 4 to a bag. How many bags can you fill from 28 cookies? This is division too, but the question changed. You know the size of each group, 4. You want the number of groups. Take away 4 at a time: 28, 24, 20, 16, 12, 8, 4, 0. That fills 7 bags. Again, 28 ÷ 4 = 7.

This is division as grouping. Sharing asks how many are in each group. Grouping asks how many groups there are. Both use the same fact, 4 × 7 = 28. The story decides which question you are answering. Twenty cups, 5 to a tray: 20 ÷ 5 = 4 trays. Twenty cups spread over 5 trays: 20 ÷ 5 = 4 cups on each tray.

Grouping is repeated subtraction, just as multiplying is repeated addition. To find 24 ÷ 6, subtract 6 until you reach 0: 24, 18, 12, 6, 0. You subtracted 4 times, so 24 ÷ 6 = 4. Count the subtractions, not the numbers you wrote. A student who counts the list 24, 18, 12, 6, 0 and says 5 has counted the starting number.

Division has its own names. In 28 ÷ 4 = 7, the number being divided, 28, is the dividend. The number you divide by, 4, is the divisor. The answer, 7, is the quotient. The dividend is the whole pile. The divisor is how you split it. The quotient is what you get.

Order matters in division. 28 ÷ 4 = 7, but 4 ÷ 28 is a tiny piece, not 7. Always write the total first. Some problems say divide 28 by 4, and some say 4 goes into 28. Both mean 28 ÷ 4. Division is also written with a bar, like 28/4, which is the same shape you will see in fractions.

Two special divisors are 1 and the number itself. Any number divided by 1 is itself: 9 ÷ 1 = 9, since one group holds everything. Any number divided by itself is 1: 9 ÷ 9 = 1, one in each of nine groups. Zero divided by any number is 0: 0 ÷ 6 = 0, since there is nothing to share. Dividing by 0 has no answer, so we never write it.

Look at ? × 4 = 28. The box asks for a missing factor: what number times 4 makes 28? You can count by 4s until you hit 28: 4, 8, 12, 16, 20, 24, 28. That is 7 fours. Or you can recall 7 × 4 = 28. Either way, the missing factor is 7. And 28 ÷ 4 = 7 says the very same thing.

Division and multiplication are inverse operations. One undoes the other. Multiplying 7 by 4 gives 28. Dividing 28 by 4 brings you back to 7. So every division fact is hiding inside a multiplication fact you already know. To find 45 ÷ 9, think 9 × ? = 45. Since 9 × 5 = 45, the answer is 5.

The missing factor can be in either spot. 6 × ? = 48 and ? × 6 = 48 have the same answer, 8, because of the commutative property. A common mistake is to solve 6 × ? = 48 by adding: 6 + 42 = 48, so ? = 42. Check by multiplying: 6 × 42 is far more than 48. The answer 6 × 8 = 48 is right.

The numbers 4, 7 and 28 are related. 4 × 7 = 28. 7 × 4 = 28. 28 ÷ 4 = 7. 28 ÷ 7 = 4. These are related facts. Together they form a fact family. They share the same three numbers. The product, 28, is always the biggest. In the divisions, the biggest number comes first. Learn one fact and you get all four.

Write the family for 6, 8 and 48. Multiplications: 6 × 8 = 48 and 8 × 6 = 48. Divisions: 48 ÷ 6 = 8 and 48 ÷ 8 = 6. Notice that 48 ÷ 6 = 8 and 48 ÷ 8 = 6 are different facts. Both are true. But you cannot switch the numbers in a division; 6 ÷ 48 = 8 is false.

Fact families make division fast. If you see 63 ÷ 7 and freeze, ask which family 63 belongs to. 7 × 9 = 63, so the family is 7, 9 and 63. That means 63 ÷ 7 = 9. When two numbers in a family are the same, like 5, 5 and 25, the family has only two facts: 5 × 5 = 25 and 25 ÷ 5 = 5.

The multiplication table can divide, too. To find 42 ÷ 6, go to the row for 6. Slide across until you find 42. Look up to the top of that column. It says 7. So 42 ÷ 6 = 7. You just found the missing factor in 6 × ? = 42 by looking instead of guessing.

Try 72 ÷ 8. The row for 8 reads 8, 16, 24, 32, 40, 48, 56, 64, 72. The 72 sits in the ninth column, so 72 ÷ 8 = 9. If the dividend is not in the row, the division does not come out even. Look for 50 in the row for 8 and you will not find it. Something will be left over, which is the next lesson.

Use the table to build speed, then let it go. Cover the table and try 36 ÷ 4. Think: 4 × ? = 36. Since 4 × 9 = 36, the answer is 9. Try 81 ÷ 9: 9 × 9 = 81, so 9. The goal is knowing the facts by heart. A student who reads 42 ÷ 6 from the row for 7 will get 6, which is a true fact from the same family, but not the one asked.

Four friends share 30 cookies. Deal them out: 7 each uses 28 cookies. Two cookies are left, and 2 is not enough for another round of 4. So 30 ÷ 4 = 7 with a remainder of 2. We write 30 ÷ 4 = 7 R 2. The remainder is the part that does not fit into the equal groups.

To divide with a remainder, find the biggest multiple of the divisor that fits. For 23 ÷ 5, count by 5s: 5, 10, 15, 20. The next one, 25, is too big. So 4 fives fit, using 20. Then 23 − 20 = 3. So 23 ÷ 5 = 4 R 3. Check: 4 × 5 = 20, and 20 + 3 = 23.

The remainder must be smaller than the divisor. If it is not, another group still fits. A student who writes 23 ÷ 5 = 3 R 8 has stopped too early, since 8 is bigger than 5. One more five fits, giving 4 R 3. Always check with quotient × divisor + remainder = dividend.

A remainder is not always the final answer. Read the story again. Vans hold 6 kids, and 27 kids are going to the museum. How many vans? 27 ÷ 6 = 4 R 3. Four vans carry 24 kids, and 3 kids would be left behind. They need a van too. So the answer is 5 vans. Here you round up.

Sometimes the leftover is dropped. Bracelets need 6 beads each, and Lin has 27 beads. How many bracelets can she make? 27 ÷ 6 = 4 R 3. She makes 4 bracelets. Three beads are not enough for another one. Here the remainder just sits in the bag. The answer is 4.

Sometimes the remainder itself is the answer. Lin makes her 4 bracelets. How many beads are left over? The answer is 3. And sometimes you share the leftover: 3 cookies for 4 friends means each gets a piece, which is a fraction. Ask what the question wants: the groups, the leftover, or the amount each. Then decide.

Mia buys 3 packs of 8 pencils and gives 5 to her brother. How many does she keep? No single operation answers this. Take it one step at a time. Step 1: find the pencils. 3 × 8 = 24. Step 2: take away the gift. 24 − 5 = 19. Mia keeps 19 pencils. A two-step problem needs the answer from step 1 to do step 2.

Read the whole problem before you start. Ask: what do I need to know first? A class of 24 splits into teams of 4, and each team gets 2 balls. How many balls? First find the teams: 24 ÷ 4 = 6. Then the balls: 6 × 2 = 12 balls. If you did 24 × 2 first, you would count balls for every student instead of every team.

Write a short number sentence for each step. Label the answer of step 1 so you do not lose it. Then check the final answer against the story. Twelve balls for 6 teams sounds right. A common mistake is to stop after step 1 and answer 6, the number of teams, when the question asked for balls.

Look at 2 + 3 × 4. Two students get different answers. One adds first: 2 + 3 = 5, then 5 × 4 = 20. The other multiplies first: 3 × 4 = 12, then 2 + 12 = 14. Math needs one answer, so there is a rule called the order of operations. Multiply and divide before you add and subtract. The right answer is 14.

The marks ( ) are parentheses. They change the order and say do me first. So (2 + 3) × 4 means add first: 5 × 4 = 20. Without parentheses, 2 + 3 × 4 = 14. Both are correct sentences; they just say different things. Try 20 − 8 ÷ 2. Divide first: 8 ÷ 2 = 4, then 20 − 4 = 16.

Use parentheses to write a two-step problem as one sentence. Mia’s pencils: 3 × 8 − 5 = 24 − 5 = 19. The team balls: (24 ÷ 4) × 2 = 6 × 2 = 12. Here the parentheses are not needed, but they show your plan. When a sentence has only multiplying and dividing, work from left to right: 24 ÷ 4 × 2 = 6 × 2 = 12.

An estimate is a quick, close answer. Before you solve 4 × 19, think: 19 is close to 20, and 4 × 20 = 80. So the answer should be a little under 80. Now solve: 4 × 19 = 76. That is close to 80, so it makes sense. If you had gotten 46, the estimate would tell you to look again.

To estimate, round each number to a friendly one, like a ten. For 6 × 28, round 28 to 30: 6 × 30 = 180. The exact answer, 168, is near 180. For division, pick a nearby fact. To check 63 ÷ 7, think 63 is near 70, and 70 ÷ 7 = 10. So 63 ÷ 7 should be a bit less than 10. It is 9.

Estimating also checks whether an answer is sensible. Each of 8 classes has 23 students. Are there about 40 students or about 160? 8 × 20 = 160, so about 160. A student who says 31 added 8 + 23. Use the estimate to catch that kind of slip. An estimate is not the answer; it is the guard that protects your answer.