Finding the missing part

1.

Pose this word problem, or an adaptation that reflects a context that is relevant to your students.

• You take 20 shots at goal and score with 55% of the shots.
• How many shots do you score with?

Let students attempt the problem, possibly in pairs.

Students may solve the problem using trial and error. Since 55% is more than one half, they may try fractions like 12/20 or 11/20 to find out if those fractions equal 55%. The approach is likely to be successful in this example.

Use a double number line to organise the information from the problem and display their solution strategies.

An alternative is to consider:

• What percentage is each shot worth? How do you know? (5% since 100% divided by 20 equals 5%)
• How many shots, at 5% each, make 55%? (11 since 11 x 5 = 55%)

2.

Pose a more difficult problem that allows for less reliance on trial and error strategies. Allow students to work in groupings that will encourage peer scaffolding and extension, as well as productive learning conversations. Consider your students' fraction and multiplication basic facts knowledge when setting these problems. You might also introduce relevant te reo Māori kupu, such as ōrau (percent). 

Good examples might include:

• You take 30 shots at goal and score with 60% of the shots. How many shots do you score with?
• What estimate is good for the number of goals? Since 60% is more than one half, an estimate above 15 goals is reasonable.
• What fraction is 60%? Can you use that knowledge to solve the problem? (60% = 2/5 or 6/10)
• How many percent is each goal worth? How can we work that out? (100 ÷ 30 = 3.3 or 3 1/3).
• What percentage is three goals worth? What percentage is six goals worth?

Confirm the answer, 18 goals, and work backwards to check it is correct.

Use a calculator: 18 ÷ 30 % = 60%

Ask students to draw a double number line that represents the important information.

3.

Pose increasingly difficult problems and support students to rely more on generalisation strategies and less on trial and error. Examples might include:

a.

• You take 40 shots at goal and score with 65% of the shots. How many shots do you score with?
• Each shot is worth 2.5% since 100 ÷ 40 = 2.5. 65 ÷ 2.5 = 26, so 26 shots were goals.
• Check: 26 ÷ 40% = 65%.

b.

• You take 25 shots at goal and score with 64% of the shots. How many shots do you score with?
• Each shot is worth 4% since 100 ÷ 25 = 4. 64 ÷ 4 = 16, so 16 shots were goals.
• Check: 16 ÷ 25% = 64%.

c.

• You take 80 shots at goal and score with 95% of the shots. How many shots do you score with?
• Each shot is worth 1.25% since 100 ÷ 80 = 1.25. 95 ÷ 1.25 = 76, so 76 shots were goals.
• Check: 76 ÷ 80% = 95%.

1.

Increase the level of abstraction with the aim of having students use symbolic form. Support students in using empty number lines and equations to organise and record the information. Use examples in which the denominator is not a tidy factor of 100. For example:

• You take 48 shots at goal and score with nearly 73% of the shots.
• How many shots do you score with?
• Each shot is worth about 2.08% since 100 ÷ 48 = 2.08333...
• 73 ÷ 2.08 = 35.096... so 35 shots were goals.
• Check: 35 ÷ 48% = 72.9166...%