start unknown

Addition and Subtraction Part 5: Separate and Comparison Problem Solving Structures KG-4th

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by C. Elkins, OK Math and Reading Lady

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.

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Addition and Subtraction Part 4: The Equal Sign and Join Problem Solving Structures KG-4th

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by C. Elkins, OK Math and Reading Lady

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).

Helpful materials to teach and practice these strategies are bar models, part whole templates, a balance scale, and ten frames using cubes and/or two color counters. Here are 2 free PDF attachments. First one: Addition and Subtraction Story Structure Information. Second one is a copy of each of the 4 anchor charts shown below:  Join and part-part-whole story structure anchor charts

Join: Result Unknown

  • 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?”

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