Step-by-Step Process for Model Drawing

Read the problem. Rewrite the question in the problem as a complete sentence, leaving space

for the answer. Identify the who and the what. Draw a unit for each variable. Chunk the problem, and adjust the unit bars to match the information. Fill in

the question mark. Work your computation, and solve the problem correctly. Write the answer in the sentence.

*As you've probably noticed, model drawing sets up problems a little differently than we're used to, with horizontal writing instead of vertical math. That's a key difference, one that you and students will come to appreciate. So here's what we do.

Setting Up Our Problem

Take a horizontal sheet of paper, and make sure the problem is written at the top of it. Then write your variables on the far left side of the page. As you determine a variable in the problem, you may circle or underline it in the word problem itself. Later on, this can remind you of your variables. But that's just an optional step—if you like it, use it. And if you have visual students, encourage them to use it, too.

If there are multiple variables, write each in the order in which it appears in the problem. For example, if the problem mentions Mike first and then Elisha, put Mike before Elisha as you list variables one under the other.

**Now we're going to solve three kinds of addition word problems, from simple to complex. Addition problems are a great way for any student to get his or her bearings in model drawing. The problems don't get too complicated on us, and as a result, much of the math is simple. This lets kids focus on learning the process of model drawing rather than the content (the correct answer for one problem).

So even if you have fifth-graders who are new to model drawing, addition problems offer just the right introduction.

 Problem 1 (Part Whole Model)

Sean has 2 plastic cars. He also has 5 wooden cars. How many cars does Sean have altogether?

Problem 3

Jesse had $3.00 more than Clinton. If Clinton had $10.00, how much money did they have altogether?

Every word problem usually falls under the category of a part-whole or comparison. After we read the problem, and before we set up our model, it's a good idea to spend some time talking about what kind of model we want to use. I ask my students questions such as these to help guide their thinking:

Are we solving for the part or the whole?

Do we know how many parts are involved in this problem?

Does each part have the same amount?

Are we comparing values at all?

What are we solving for?

Problem 5 (Before and After Addition Problem)

John had 10 pencils, Andrew had 9 pencils, and Calvin had 5 pencils. They decided they'd put their pencils together and share them equally. How many pencils did each student get?

SUBTRACTION

Problem 1 (Part-Whole Model)

Nathan had $27.00 to buy gifts for his family. If he spent $9.00 on a gift for his brother, how much money did he have left to spend on the rest of his family?

Problem 3 (mixed operations)

Ryan and Chris started out with an equal amount of baseball cards. Ryan lost 15 cards, and Chris collected another 45 cards. How many more cards did Chris have in the end?

Problem 5 (multi-step subtraction)

The pep club made 425 buttons to sell on Friday. The club sold 75 more buttons in the morning than they did in the afternoon. If all the buttons were sold, how many buttons did the pep club sell in the morning?

Multiplication

Before we get going, let me share a couple of golden rules of multiplication model drawing. These will help you today, and they'll help your students tomorrow.

When a problem says, "There were ____ times as many," hone in on what that means. Add one unit to your unit bar at a time, and count with students while adding each unit. (I call this the counting method. For example, if you have, "There were 4 times as many," say, "Okay, now we start with 1 times as many because our models are equal. Now we add 2 times as many (one unit bar), 3 times as many (another unit bar), and 4 times as many (a third unit bar)" so they can see how each unit makes an "as many."

If we don't do it this way, students might see "4 times as many" and think they add 4 units to the base unit bar. That will skew the answer.

For multiplication problems, it's usually helpful to draw a smaller unit bar to begin with. That way, you can add to it.

The computation is where you can differentiate model drawing, allowing students to get to their answers in whatever way is easiest for them.

Check to make sure that the model mirrors the sentence. This becomes particularly important as you get into more-complex problems.

Problem 1

Ryan had 4 times as many marbles as Jordan. If they had 60 marbles altogether, how many marbles did Ryan have?

Solving More Mixed-Operation Multiplication Problems

Problem 3

A ribbon that's 1,530 inches long is cut into two pieces. The length of one piece is 2 times the length of the other. What is the length of the longer piece?

Mixed-Operations Multiplication Problems With Three Whos

Problem 5

Mike had 3 times as much money as Steve. Dick had 2 times as much money as Steve. If Dick had $98.00, how much money did Mike have?

DIVISION

But before we get ahead of ourselves, let me share two really important secrets about division problems.

With these problems, we show our units by dividing one long unit bar into its parts. So for example, if we have one unit that's equal to 10, and the problem asks us to divide it in half, we'll draw a vertical line through the center to show the halves.

If the question asks for the value of one of the parts of a unit bar, go ahead and place your question mark right inside that section of the bar.

Problem 1

28 chairs are arranged equally in 4 rows. How many chairs are there in each row?

Part-Whole Division Word Problems

Problem 3

A man divided his fortune of $92,000 into 8 equal parts. He gave 4 portions to his wife, 1 portion to his son, and divided the rest equally among three charities. How much more money did the wife receive than the son?

With problems like these, we don't need to draw separate unit bars for each "piece holder." On the contrary, we have to block out that information temporarily so we can figure out how the parts relate to their whole.

In this problem, what we really need to know is what a part of the man's fortune equals.

That means our who is the man.

Problem 5

Eric had 3 times as many cookies as Sean. After Eric ate 50 cookies, he had half as many cookies as Sean. How many cookies did Eric have left?

I notice that the word after is in my word problem. This can trigger me to remember that I might want to set up two models (a before model and an after model), because my model will change.

If you find your students struggling to remember when to draw long and short unit bars, it helps to point out that the positive operations (addition and multiplication) require short bars, and the negative operations (subtraction and division) require longer ones. In those negative operations, we often do work inside the bars, so it's helpful to have some extra room.

Fraction Word Problems Fraction problems offer us a unique choice in the way we set up unit bars. One way of

doing it is by drawing a long unit bar (like we do in division) and sectioning it off to reflect fractional parts. The other way is to identify the variables (the parts of a whole) and draw separate unit bars for each.

See what I mean? These choices are great, and they give students the opportunity to do what makes the most sense to them, but they can also cause confusion. The message here is, "Offer a little guidance and a lot of practice!"

When you see the words remainder or remaining in a fraction word problem, draw a long unit bar because most likely you'll be carrying down the remaining part into a new unit bar set. Here's an example problem with a remainder. At ABC Cookie Company, 3/5 of the 300 bakers made sugar cookies. 2/3 of the remaining bakers made gingersnap cookies. If the rest made chocolate chip, how many people made chocolate chip cookies?

Here we'll need a long unit bar that we can divide into our fractional piece

Problem 1

In the 4th grade, 3/7 of the students are boys. If there are 36 girls in 4th grade, how

many students are there altogether? Ans 63

Fraction Problems With Decimals

Problem 3

After spending 1/4 of his money, Frankie had $13.50 left. How much money did he have at first? Ans. 18

Rate Word Problems With these problems, we need to use a double label because we're comparing units (like

miles to hours it takes to travel them or baskets to the minutes it takes to shoot them). One value will go inside the unit, and the other one will go outside the unit, and we do this where it makes the most sense for each problem. We have to be careful to label the values neatly so we can keep them all straight.

We can use small or large unit bars, depending on the problem's information.

Problem 3

A green car traveled 360 miles on 6 gallons of gas. How far could that same car travel on 10 gallons of gas?

Problem 5

A pool is filled with water at the rate of 75 gallons every 6 minutes. How long will it take for the pool to fill with 375 gallons of water?

Ratio Word Problems As always, we're going to go from solving simple ratio problems to more-complex ones, but before we get too far along, let me tell you three golden rules of ratio model drawing.

Have the students use manipulatives to build their model before they draw it. Since a ratio is a comparison of two or more rates, your goal is to find the base unit. Generally, we want to keep the unit bars smaller to begin with. That way, we can easily

add to them.

Problem 1

The ratio of peanut butter bars to chocolate bars to caramel bars was 2:1:3. If there were 12 chocolate bars, how many caramel bars were there?

 Problem 3

Susan and Apple shared comic books in the ratio of 3 to 5. If Apple had 14 more comic books than Susan, how many comic books did the girls have in all?

Problem 5

The ratio of Austin's money to Drew's money was 5:3 at first. After Austin spent $18.00, they had a ratio of 2:3. How much money did they have altogether at first?

PERCENT PROBLEMS

We divide our unit bars into percent rulers, which are long unit bars that can accommodate 100% (or other quantities, depending on the problem) divided up in different ways.

Percent problems are similar to fraction problems in that you can make either one percent ruler that you divide into parts or multiple percent rulers, one for each quantity represented. For example, if you know that there are 30% boys and 70% girls and a total of 220 students, you could draw the unit bars either way.

Both examples are correct. Throughout the lesson, we'll solve our problems in a variety of ways so you can see a number of examples. But if you get stuck at any point when you're working on the word problem challenges, assignment, or tests, just remember that you can always use the kinds of unit bars that make the most sense to you. There's no right or wrong!

Problem 1

At the festival, there are 210 kids. 70% of them are girls. How many boys are at the festival?

Problem 3

There were 140 students who took part in the math competition. If 90% of the students got blue ribbons, how many students got blue ribbons?

Problem 5

Toby has 75% as many coloring books as Mitch. Jarrett has 25% more coloring books than Mitch. If Toby has 45 coloring books, how many coloring books does Jarrett have?

 Problem 1

Wren and Lily had bunches of flowers to sell at the local Farmer's Market in the ratio of 3 to 4. If Lily had 16 bunches of flowers to sell, how many bunches of flowers did they have to sell in all?

 Problem 2

John, Kelly, and Marie went surfing at the beach. They drove in separate cars. It took John 20 minutes to get to the beach. It took Kelly 15 minutes more than John to get to the beach. Marie made it to the beach 10 minutes earlier than Kelly. How long did it take Marie to get to the beach?

 Problem 3

Jesse had friends over to celebrate their school's football victory. They ordered pizza. If there were 17 kids at the party and each one received three pieces of pizza, how many pieces of pizza did they eat in all?

Problem 4

Marie spent 3/5 of her money on books. She had $75 to begin with. How much money did Marie spend on books?

 Problem 5

Connor collected 3 times as many shells as Nadine. If Connor collected 18 more shells than Nadine, how many shells did they collect in all?

Problem 6

Zandar ran 5 miles a day on Mon, Tues. and Wed. If he ran a total of 29 miles in 5 days, how many miles did he run in the last two days?

 Problem 7

Davis has 82 trading cards. Aiden has 12 less than Davis. The number of cards owned by Davis, Aiden and Clara adds up to 214. How many fewer trading cards does Clara have than Aiden?

 Problem 8

Nana June made 30 cupcakes. She needs to place them in boxes. Each box will hold 6 cupcakes. How many boxes will Nana June need?

 Problem 9

Mitchell and Winsome went out to dinner. The bill totaled $80. Mitchell paid $15 more than Winsome. How much money did Mitchell pay for dinner?

Problem 10

Leann participated in a triathlon. She spent 1/3 of her time in the running event and 5/8 of the remainder of her time in the swimming event. If it took her 30 minutes to finish the swimming event, how long did it take her to finish the running event?