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.
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.
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.