Answers to questions about counting on.

1. Describe the process of counting on from first for the problem:

Mark has 3 Bionicles and 6 Hero Factory robots. How many robots does he have in all?

Say 3 (pause), 4 (raise 1 finger), 5 (raise a second finger), 6 (raise a third finger), 7 (raise a fourth finger), 8 (raise a fifth finger), 9 (raise a sixth finger). The answer is 9.

2. Describe the process of counting on from larger for the problem:

Mark has 3 Bionicles and 6 Hero Factory robots. How many robots does he have in all?

Say 6 (pause), 7 (raise a finger), 8 (raise a second finger), 9 (raise a third finger). The answer is 9.

3. What CGI problem type is most closely associated with the counting on strategy?

Join, Result Unknown.

4. For what addends is counting on an efficient computation strategy?

Counting on is efficient for problems where one addend is small (1, 2, or 3)

5. How can you get children who are in the habit of counting all to break that habit and start solving problems by counting on?

Ask problems where the first addend is large enough that counting all would be unreasonably difficult (above 20). It's also helpful to use visuals such as the Bears-in-a-cave activity, where one of the numbers (preferably the larger one) is hidden so that it's more difficult and less natural to count it from 1.

Questions to answer about counting back

6. Which CGI word problem type is most closely associated with the counting back strategy?

SRU--Separate, result unknown

7. For what sorts of numbers is counting back an efficient strategy? (Efficient means it can be reliably be used in a short amount of time--3 seconds of less).

Problems where the subtrahend is 1 or 2

8. Describe both ways of using counting back to solve the problem: 8-3

Way number 1:

8...
7 (put up a finger)
6 (put up a second finger)
5 (put up a third finger)
The answer is 5

Way number 2:

8 (put up a finger)
7 (put up a second finger)
6 (put up a third finger)
The answer is 5.

9 How can you use manipulatives to help children understand counting back as a strategy?

Use manipulatives to model a problem where part of the manipulatives (the difference) are hidden and can't be directely counted.

For example:

Hide a given number of counters (for example, hide 8 counters under a cup and tell the children that there are 8 under the cup)
Ask children to think quietly about how many will be left under the cup if you take out two.
Take out two counters one at a time.  Ask the children how they figured out the answer.

Questions to answer about counting up to subtract

10. Which CGI word problem type is most closely associated with the counting up to strategy?

JCU

11. For what sorts of numbers is "counting up to" an efficient strategy?

Numbers where the minuend and the subtrahend are close (with a difference of 1, 2 or maybe 3)

12. Rewrite the missing number addition problem as a subtraction problem: 8 + ? = 11

11 - 8 = ?

13. Describe how to use "counting up to" to solve 7 - 5

5... 6 (put up a finger), 7 (put up another finger).  Count the fingers up, the answer is 2.

14. Why do we want children to learn to use "counting up to" to solve subtraction problems?

• They are efficient for a new set of subtraction problems (ones where the minuend and subtrahend are close).
• Counting up is easier than counting back
• It reinforces the relationship between addition and subtraction.

15. What would be a good problem type to compare counting back and counting up to?

PPW-PU  (CDU would also be an acceptable answer).