The purpose of this activity is to engage students in reasoning with unknowns and finding plausible values for the unknowns using information provided. The information can be expressed as three equations involving three unknowns. Students might also apply numeric processes involving addition, and multiplication.
The background knowledge presumed for this task is outlined in the diagram below:
This activity should be used in a ‘free exploration’ way with an expectation that students will justify the solutions that they find.
The procedural approach (show more)
- The student uses estimation and appropriate calculations, to find the value of the three unknowns by systematic trial and improvement.
In a procedural approach, students’ strategies are likely to be arithmetic. They might use trial and error at first by speculating on the weight of the three athletes and carrying out calculations to check. This is a cumbersome approach that needs refinement to be successful.Click on the image to enlarge it. Click again to close.
The zone of possible values is reduced considerably if students notice features in the graphics. First, they might reason about the differences in weights. Since the weight of the top person is always equally shared, the first picture shows that Carol is 87 – 79 = 8 kg heavier than Bart. The third picture shows that Bart is 88 – 78 = 10 kg heavier than Anna.
Second, considering the combined weight all three people is always 166 kg this gives an average weight of about 166 ÷ 3 ≈ 55 kg. Using the differences and average will produce estimations that are potentially more fruitful.Click on the image to enlarge it. Click again to close.
The conceptual approach (show more)
- The student uses algebraic equations, or graphs, to find three unknowns from three linear equations.
Simultaneous equations are learned at Levels 6 and 7 of the New Zealand Curriculum. Students need to connect the information across pictures to solve for the unknowns. Using the information from one picture is likely to lead to this scenario.Click on the image to enlarge it. Click again to close.
Connecting information across pictures allows for elimination solutions like this:Click on the image to enlarge it. Click again to close.