As I promised, here is a post about another multiplication and division story structure.
The dinosaur is about twice as tall as the human. The dinosaur is about 12 times wider than the human.
While the previous structure I mentioned dealt with equal groups, this one deals with comparisons. Like with addition / subtraction comparison problems, I will show some ways to use double bars to illustrate these. (I just added a FREEBIE – a 2-pg. pdf showing the pictures of the problems shown below – get it now Equal groups comparison problem pics)
Notice some of the questions associated with these stories. They sound a lot like questions used in other story structures — this is why I advise against statements like, “If it asks how many in all, add the numbers together.” Notice how this method guides a student as they go from pictorial to abstract. You can also represent these types of problems with cubes or money manipulatives if you need a concrete example.
This problem shows a multiplying process.
Read the problem. Ask who and what this story is about (Joe and Brent – Joe has more $ than Brent). Notice there is no additive or subtractive process, but one has more – the other has less. This is a signal to use comparison bars to help solve.
Make a vertical line (to help line up the bars evenly on the left side. Draw the bar with the known amount. (Joe has $22.) Label the second bar (Brent).
3. Since the story said Brent has 3 time more, draw 3 bars equal to the size of Joe’s bars.
4. Since Joe’s bar is $22 and the bars by Brent’s name are equal in size, all of Brent’s bars should be labeled with $22.
5. To find out how much Brent has, solve by repeated addition or multiplication.
6. IF the question was slightly different (How much do Joe and Brent have all together?), then the above steps would have to be followed by 1 more step (adding the two amounts together).
* If students had been taught earlier to “Add when the question asks How many in all” then the child would likely add $22 + 3 to get their answer. That signals no conceptual understanding of what the problem is all about: One has more, the other has less.
Are you ready for a way to help students solve multiplication and division word problems? I have developed a template that will help students record the given facts and think through the process involving equal groups. There are 3 basic types of equal groups problems:
# of groups and # in each group are known
# of groups and total are known
# in each group and total are known
Pictured here is an illustration of the strategy which can be used for multiplication and division problems. Get my FREE packet right here: Equal groups strategy with template.
One of the most important steps I recommend when working with equal groups problems is for students to brainstorm things that typically come in equal groups. There is a great book titled “What Comes in 2’s, 3’s, and 4’s” which can set the stage. It’s a picture book, but helpful to get kids’ brains warmed up. Here are some visuals I made to illustrate the point. Click here for your FREE copy. They are included in this set of Equal groups pictures and list template
Kids may need your help to think of things to add to the list. See some hints below. This template and a full list is included in the above FREE download. Continue reading →
Solving comparison problems involves a different thinking process than addition or subtraction problems. In my previous posts Math Problem Solving Parts 1 and 2, I focused on addition and subtraction problems in which I advised students to think of the verbs used (found, bought, earned, spent, lost, gave away, etc.) to help visualize what is going on in the story. In comparison problems, there are no additive or subtractive processes present – just analyzing which has more, which has less, and how much more or less one amount is compared to another.
I love using a double bar method for comparison problems. It can be represented with manipulatives for a concrete experience (ex: connecting cubes, tiles) or with a drawing of two rectangular bars for a pictorial representation. These two visual models provide students with the concept of comparison — they can see right away which has more, which has less. Then the information can be used to compare the two amounts. Look for the FREE resource at the end of this post.
I recommend the use of manipulatives when dealing with students’ first experience with comparison problems. Dealing with quantities less than 20 make using manipulatives manageable. I will show this in a horizontal format, but it can certainly be done vertically. You can also apply this quite well to graphing problems.
Problem: I have 12 crayons. My friend has 8 crayons. How many more crayons do I have than my friend?
Determine that this problem is not an addition / subtraction process (because no one is adding to what they have and no one is giving away what they have), but in fact a comparison problem.
Be cautious about focusing on the question”how many more” and telling students that when they see this they need to subtract. When we tell kids to focus on the specific words in a question with a “rule” for adding or subtracting, they start to lose sight of the actual story.
Consider that this question can also be used with a SSM (some and some more) story which is not a comparison problem: What if it read: “I have 8 nickels. I want to buy a candy bar that costs 80 cents. How much more money do I need?” This story means: I have some (40 cents) and I need some more (?) so that I have a total of 80 cents. This is a missing addend or change unknown story: 40 + ___ = 80. Even though it could be solved in a similar manner as a comparison story, we want students to be able to tell the difference in the types of stories they are solving.
Determine who has more (represented by yellow tiles), who has less (green tiles).
The crayon problem can be solved by lining up 2 rows of manipulatives (pictured):
Notice the extras from the longer bar. Count them (4)., or
Count up from 8 to 12 to find the difference.
Even if the question was “How many fewer crayons does my friend have?” it would be solved the same way.
With pictorial double bars:
Problem Type 1 (Both totals known): Team A scored 85 points. Team B scored 68 points. How many more points did Team A score than Team B?
Determine this problem is not an addition / subtraction process (because the teams are not gaining or losing points).
Ask “Who?” and “What?” this story is about: Team A and B and their scores.
Draw double bars (one longer, one shorter) which line up together on the left side.
Label each bar (Team A, Team B).
For the team with the larger amount (Team A), place the total outside the bar (85).
For the team with the smaller amount (Team B), place the total inside the bar (68).
Make a dotted line which extends from the end of the shorter bar upward into the longer bar.
Put a ? inside the extended part of the longer bar. This is what you are trying to find.
To solve, there are 2 choices:
68 + ____ = 85 This choice might be preferred for those with experience using mental math or open number lines to count up.
In the previous post, I addressed problems dealing with an additive process (join; aka SSM). In this post, I will show you some models to use for these types of problems: Separate; aka Some, Some Went Away — SSWA. I will share some models which are great for KG stories as well as templates that are helpful for intermediate students to use, especially when dealing with missing addend types.
As I previously mentioned, it is my belief that students should focus more on the verb / action in the story and not so much with the key words we often tell kids to pay attention to. Brainstorm actions that signify a subtractive process. Post it in the class. Keep adding to it as more actions are discovered. With this subtractive action, kids should quickly realize that there should be less than we started with when we take something away. Here is a FREE poster showing some of the most common subtraction action verbs. Click HERE to get your copy.
I have 1 penny in one pocket. I have 6 more pennies in another pocket. This is a Join or “some and some more” story structure.
Many teachers I work with have asked for advice on problem solving in math. Students often don’t know how to approach them or know what operation to use. Should teachers help students focus on key words or not? What about the strategies such as CUBES, draw pictures, make a list, guess and check, work backwards, find a pattern?
While all of those strategies definitely have their purpose, I find we often give kids so many steps to follow (underline this, circle this, highlight that, etc.) that they lose sight of what the problem is basically about.
In this post, I will focus on two basic questions (who and what) and a simple graphic organizer that will help students think about (and visualize) the actions in a one-step Join story problem. KG and first grade students can act out these actions using story mats or ten frames. Late first through 5th grade can use a part-part-whole box. There are two FREE items offered.
These are the types of problems I will focus on in the next few posts.
Join (also referred to as SSM – Some and Some More)
Separate (also referred to as SSWA – Some, Some Went Away)
JOIN problems have 3 versions:
a + b = ___ (The result is unknown.)
a + ____ = c (How the story changed is unknown / missing addend.)
____ + b = c (The start is unknown / missing addend.)
They can also be referred to as “Some and Some More” stories (SSM). This means, you have some and you get some more for a total amount. These present themselves as additive stories, but it doesn’t necessarily mean you have to add to solve them. The second and third version above are often referred to as missing addend problems. Continue reading →
Is division a dreaded topic on your list of objectives to teach? Like many math topics, students have a harder time understanding it most likely because it’s not something they use regularly in their lives. Students should understand why division is useful before they have to start solving division problems. In this post, I will focus on helping students see the relationship between subtraction, multiplication, and division both with concrete objects, pictures, and the partial quotients model. Freebies available below!!
425 divided by 8 picture form
425 divided by 8 partial quotients model
Then let’s talk about what division really is — it is repeated subtraction; much the way multiplication is repeated addition. The issue is that repeated subtraction is not always very efficient. Here’s what I mean.
Let’s say I have the basic problem 25 ÷ 5. I could start with 25 and then subtract 5, subtract another 5, another 5, another 5, and another 5 until I run out and reach zero. I would have to do this 5 times. If I had 25 cookies that I wanted to share equally among 5 friends, I could do the “one for you, one for you, one for you, one for you, and one for you” process and still end up with 5 cookies for each. Or I could try “two for you, two for you,” etc. to make the action of passing out the cookies faster. When I get down to 5 cookies, I return to the “one for you . . .” to make it work.
With a larger problem such as 72 ÷ 6, I can again try subtracting 6 at a time until I reach zero. This would take 12 repetitions — not efficient, but still accurate. Could I subtract 12 at a time instead (2 groups of 6) to be more efficient? Or 18 at a time, or 24 at at time? This is the idea behind the partial quotients model I will refer to a little later. Continue reading →
This week I will focus on subtraction problem structures. There are two types: separate and compare. I suggest teaching these models separately. Also, some part-part-whole problems can be solved using subtraction. I will refer to the same terms as in addition: start, change, result. You can also use the same materials used with addition problems: part-part-whole templates, bar models, ten frames, two-color counters, number lines, and connecting cubes.
The goal is for students to see that subtraction has different models (separate vs. comparison) and an inverse relationship with addition — we can compose as well as decompose those numbers. Knowledge of number bonds will support the addition / subtraction relationship. Here is the same freebie I offered last week you can download for your math files: Addition and Subtraction Story Structure Information The six color anchor charts shown below are also attached here free for your use: Subtraction structure anchor charts
Separate: Result Unknown
Example: 10 – 4 = ____; There were 10 cookies on the plate. Dad ate 4 of them. How many are left on the plate?
Explanation: The problem starts with 10. It changes when 4 of the cookies are eaten. The result in this problem is the answer to the question (how many are left on the plate).
Teaching and practice suggestions:
Ask questions such as: Do we know the start? (Yes, it is 10.) Do we know what changed? (Yes, 4 cookies were eaten so we take those away.) How many cookies are left on the plate now? (Result is 6.)
Reinforce the number bonds of 10: What goes with 4 to make 10? (6)
Draw a picture to show the starting amount. Cross out the items to symbolize removal.
Show the problem in this order also: ____ = 10 – 4. Remember the equal sign means the same as — what is on the left matches the amount on the right of the equal sign.
Ask your students this question: “What does the equal sign mean?” I venture many or most of them will say, “It’s where the answer goes,” or something to that effect. You will see them apply this understanding with the type of problem in which the answer blank comes after the equal sign such as in 5 + 6 = _____ or 14 – 9 = _____. This is the most common type of problem structure called Result Unknown. I was guilty of providing this type of equation structure a majority of the time because I didn’t know there actually were other structures, and I didn’t realize the importance of teaching students the other structures. So . . . here we go. (Be sure to look for freebies ahead.)
Teach your students the equal sign means “the same as.” Think of addition and subtraction problems as a balance scale: what’s on one side must be the same as the other side — to balance it evenly.
Let’s look at the equal sign’s role in solving the different addition / subtraction structures. I will be using the words start to represent how the problem starts, the word change to represent what happens to the starting amount (either added to or subtracted), and the result. In this post I will be addressing the join / addition models of Result Unknown, Change Unknown, Start Unknown and Part-Part-Whole. I will highlight the subtraction models (Separate and Comparison) next time. Knowing these types of structures strengthens the relationship between addition and subtraction.
Three of these addition models use the term join because that is the action taken. There are some, then some more are joined together in an additive model. One addition model, however, is the part-part-whole model. In this type there is no joining involved – only showing that part of our objects have this attribute and the other part have a different attribute (such as color, type, size, opposites).
Example: 5 + 4 = ____; A boy had 5 marbles and his friend gave him 4 more marbles. How many marbles does the boy have now?
Explanation: The boy started with 5 marbles. There was a change in the story because he got 4 more. The result in this problem is the action of adding the two together.
Teaching and practice suggestions:
Ask questions such as: Do we know the start? (Yes, it is 5.) Do we know what changed? (Yes, his friend gave him more, so we add 4.) What happens when we put these together?(Result is 9.)
Show the problem in this order also (with result blank first instead of last) : ____ = 5 + 4
Common questions: How many now? How many in all? How many all together? What is the sum?
Join: Change Unknown
Example: 5 + ____ = 9; A boy had 5 marbles. His friend gave him some more. Now he has 9 marbles. How many marbles did his friend give him? You could also call this a missing addend structure.
Explanation: This problem starts with 5. There is a change of getting some more marbles, but we don’t know how many yet. The result after the change is (9). It is very likely students will solve like this: 5 + 14 = 9. Why? Because they see the equal sign and a plus sign and perform that operation with the 2 numbers showing. Their answer of 14 is put in the blank because of their misunderstanding of the meaning of the equal sign.
Teaching and practice suggestions:
Ask questions such as: Do we know the start? (Yes, he started with 5 marbles.) Do we know what changed in the story? (Yes, his friend gave him some marbles, but we don’t know how many.) Do we know the result? (Yes, he ended up with 9 marbles.) So the mystery is “How many marbles did his friend give him?”
Count up from the start amount to the total amount. This will give you the change involved in the story.
Reinforce number bonds by asking, “What goes with 5 to make 9?” or “What is the difference between 5 and 9?”
Join: Start Unknown
Example: ____ + 4 = 9; A boy had some marbles. His friend gave him 4 more. Now he has 9. How many did he have to start with?
Explanation: The boy had some marbles to start with, but the story does not tell us. Then there is a change in the story when his friend gives him some more (4). The result is that he has 9.
Teaching and practice suggestions:
Ask questions such as: Do we know how many marbles the boy started with? (No, the story didn’t tell us.) Do we know what changed in the story? (Yes, his friend have him 4 marbles.) Do we know the result? (Yes, he had 9 marbles at the end.)
Count up from 4 to 9.
Reinforce knowledge of number bonds by asking, “What goes with 4 to make 9?”