Difference with sets to twenty

1.

Pose difference problems with numbers to twenty. Start with problems related to easy-to-learn facts. Make each addend in the problem using a separate number strip, then mask the sets.

For example:

• You have 15 red counters, and I have 10 yellow counters.
• How many more counters do you have than I have?

You might make each set on a separate number strip, then mask the sets with a length of card.

Look for students to think of 10 + 5 = 15 as a related number fact and to fluently name five as the difference.

2.

Regardless of whether the students are able to solve this problem, uncover the sets and discuss how the counters can be used to solve the problem.

• How could we find out how many more red counters there are than yellow counters?

3.

If students answered the problem fluently, encourage them to consider how difference problems might be solved with "trickier" number. This can be done by illustrating the two main strategies using the counters and number strips:

Adding additional counters to the yellow set to make the same number of red and yellow counters allows the problem to be reworded as: "How many yellow counters do I add to have the same number as red?" The equation for this calculation is 10 + [ ] = 15.

Subtracting the smaller set (i.e. the yellow counters) from the larger set (i.e. the red counters) means that only the difference remains. In this case, the number of yellow counters (10) is subtracted from the number of red counters (15), leaving only the difference of 5 counters. This problem can be reworded as if I take the smaller set of counters from the larger set, how many counters are left? The equation for this calculation is 15 - 10 = [ ] .

4.

Pose similar problems. Increase the difficulty of the basic facts that students need to use. Make the sets on number strips and mask them, or work with only the word problems, using the physical model if additional support is needed.

• You have 16 red counters, and I have 9 yellow counters.
• How many more counters do you have than I have?

Record the two possible strategies as equations:

9 + [ ] = 16
16 − 9 = [ ]

• You have 14 blue counters, and I have 8 red counters.
• How many more counters do you have than I have?

8 + [ ] = 14
14 − 8 = [ ]

5.

Vary the amount of difference students are required to find to highlight different situations in which addition can be more effective than subtraction, and vice versa. Subtraction is often more suitable to problems where the difference between two numbers is larger or where the smaller number of the two is much smaller.

Examples might be:

You have 18 yellow counters, and I have 5 blue counters.

• How many more counters do you have than I have? (Subtraction is better because the difference between the two numbers is larger and the 5 is much smaller than 18.)

You have 17 yellow counters, and I have 15 red counters.

• How many more counters do you have than I have? (Addition is better because the difference between the two numbers is very small.)

6.

Challenge the students to solve open difference problems. Ask students to work together to explore what different addition and subtraction equations can be used to represent each problem. Consider allowing them to express their mathematical thinking in different ways (e.g., written, verbal, drawn diagrams, acting out, using physical manipulatives). Te reo Māori kupu, such as tāpiri (add), tango (subtract), and huatango (difference in subtraction), could be used throughout these activities.

7.

As a class, create a set of possible answers.

For example:

• You have a set of blue counters. I have a set of red counters.
• You have seven more counters than me.
• How many counters do we each have?

This problem, involving 12 blue counters and 5 red counters, might be represented as 5 + 7 = 12 or as 12 − 7 = 5.

• You have a set of blue counters. I have a set of red counters.
• You have seven more counters than me.
• How many counters do we each have?

This problem, involving 15 blue counters and 8 red counters, might be represented as 8 + 7 = 15 or as 15 − 7 = 8.

8.

For each set of possible answers, record the equations in sequence to check for omissions. For example:

• 12 + 7 = 19 or 19 - 7 = 12; 11 + 7 = 18 or 18 - 7 = 11; 10 + 7 = 17 or 17 − 7 = 10.

9.

Ask related difference problems outside the set of facts, such as:

• You have 7 more counters than me. I have 14 counters. How many do you have?

Increase the level of abstraction by covering the materials, asking anticipatory questions, and working more with symbols than real objects.

A suggested sequence for extending the difficulty of the differences is:

1. Find differences with doubles (e.g., 12 − [ ] = 6).
2. Find differences with "ten and…" facts (e.g., 17− [ ] = 10).
3. Find differences with close to ten facts (e.g., 15 − [ ] = 6).
4. Find differences with more difficult facts within twenty (e.g., 14 − [ ] = 8).