Model and Check Multiplication and Division Equations

Some sentences name the operation for you. A product is the result of multiplying, and the numbers you multiply are its factors. A quotient is the result of dividing. The number being divided is the dividend, and the number you divide by is the divisor. A divisor is never zero, because dividing by zero has no meaning.

In sentences like the ones in the next table, the word is marks the equals sign. Order matters in division. The quotient of a number and 2.5 means the number divided by 2.5, not 2.5 divided by the number.

A story does not always name the operation. You have to find the relationship yourself. Every story in this lesson is about equal groups, which are groups that all have the same size. Such a story links three quantities:

• the number of groups,
• the size of each group,
• the total.

When nothing is left over, the number of groups times the size of each group equals the total. Turn that around and you get the division form: the total divided by the number of groups equals the size of each group. The number of groups is the divisor there, so it cannot be zero.

A story gives you two of the three quantities. The third is the unknown. Before you write an equation, say what your variable stands for and name its unit. For example: let $b$ be the number of batches, or let $L$ be the length in kilometers.

Once the equation is written, you solve it the way you already know. Keep in mind why that method is safe, because you will need the reason when you look for mistakes. The two sides of an equation name the same number. If you do the same operation to both sides, you do it to that one number, so the results are still equal. You may multiply both sides by the same number. You may divide both sides by the same number, as long as that number is not zero.

Choose the operation that undoes what was done to the variable. Dividing by 3 undoes multiplying by 3. Multiplying by 4 undoes dividing by 4. Operations that undo each other like this are called inverse operations.

A story needs two more steps after you solve. Check your value in the original equation. A value that makes the equation true is called a solution. Then say what the solution means, with its unit. Try this on the two equations from the first example.

• In $3n = 42$, the variable $n$ is multiplied by 3. Divide both sides by 3, which is not zero: $n = 42 \div 3 = 14$. Check: $3 \times 14 = 42$. Lena buys 14 notebooks.
• In $\frac{L}{4} = 2.5$, the variable $L$ is divided by 4. Multiply both sides by 4: $L = 2.5 \times 4 = 10$. Check: $\frac{10}{4} = 2.5$. The trail is 10 kilometers long.

That check came out exactly equal because the quotient was exact. Not every answer is like that, so match your check to the kind of answer you have. The next table shows four cases. In it, the symbol $\approx$ means is about equal to.

A remainder is the whole amount left over when things cannot be shared out evenly. Compare the last two rows of the table. Ribbon can be cut, so each piece is 6.25 meters and nothing is left over. Stickers cannot be cut, so each friend gets 6 stickers and 1 sticker is left over. Do not turn a leftover into a fraction of something that cannot be split. Do not invent a leftover when the quantity can be split. So always ask what your variable counts or measures before you report an answer.

Now go the other way. Start with an equation and find its story. Two stories can use the very same numbers and still mean different things. So do not match by the numbers. Match by the relationship: ask which quantity is the total, which is the number of groups, and which is the size of each group.

It also helps to know when two forms mean the same thing. An expression is a group of numbers, variables and operations with no equals sign, such as $4x$. Here $x$ can be any number. Two expressions are equivalent when they give the same value for every value the variable is allowed to take. For example, $x \times 4$ and $4x$ are equivalent, because the order of two factors never changes a product. So a story that gives $x \times 4 = 24$ matches the equation $4x = 24$.

Be careful with a single test. The expressions $4x$ and $\frac{x}{4}$ both equal $0$ when $x = 0$. One matching value does not prove that two expressions are equivalent. At $x = 8$ they give $32$ and $2$. That one mismatch is enough to prove they are not equivalent.

The last job is to be the checker. When you look for a mistake, do not just redo the problem. Go through the work one line at a time and ask three questions:

1. Does the equation match the words?
2. Does each step do the same valid operation to both sides?
3. Is the arithmetic right?

Then test the final value. If it is meant to be exact and it does not make the original equation true, something went wrong. Find the first line that does not follow from the line before it. Also ask whether the value makes sense in the story.

Here is the idea to keep. An equation is a short way to tell a story, so each quantity must sit in its right place: the total, the number of groups and the size of each group. When you solve, do the same valid operation to both sides, and never divide by zero. When you finish, send your answer back to the original equation and to the story. Those same habits let you pick the story that fits an equation and find the line where someone's work went wrong.