A day's painting

A day's painting

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This task is about reading information off a graph.
Hohepa spent a day in his holidays painting a concrete wall for his father.
He started preparing at 9 a.m., and finished painting for the day at 5 p.m.
The following graph shows his progress. 

Question 1Change answer

Area of wall painted over the day
 
graph-painting-day.png
Write a description of what happened over the day:
  • pay particular attention to the slopes of the various parts of the graph line,
  • estimate the times when these change, and
  • write what you think Hohepa is doing at each stage.
Task administration: 
This task can be completed with pencil and paper or online (with NO auto marking displayed to students).
Level:
4
Description of task: 
Students interpret time frames and slope from a graph showing the area of a painted wall over time.
Learning Progression Frameworks
This resource can provide evidence of learning associated with within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
The key elements that need to be included in the description are the timeframes, the pace (slope), and incorporation of the context.
A possible description of what the graph shows could be:
  1. Hohepa prepares the wall, scraping down moss, etc. for the first 1 1/2 hours, with a break for morning tea at the end.
  2. He starts painting slowly, but steadily, and stops for lunch at 12 noon.
  3. He begins again at 1 p.m., and paints at a faster rate till 3 p.m. Then he has a short break for afternoon tea.
  4. He continues painting at a slower rate again until knocking off at 5 p.m.
Sufficient descriptions (see Diagnostic and formative information for examples of student descriptions)
 
Complete and sufficient descriptions [difficult]
Description includes reference to the timeframe, pace, and incorporates the context.
 
Partial answers [easy]
Description includes reference to the timeframe, incorporates the context, but not the pace of work, or
Description indicates a basic understanding of the overall nature of the graph, but with an element/detail missing, e.g., does not mention the 1 1/2 hours "preparation time" at the start, or the break for afternoon tea, or makes little attempt to estimate the times of each stage.
 
Incomplete/Insufficient detail
Answer showing minimal understanding of nature of graph, but with some elements present, perhaps noting that Harry stopped painting for lunch at 12 noon, or
Answer showing little or no understanding of graph, and making no attempt to write a reasonable description.
 
Based on a sample of 40 Y7-8 students.
Diagnostic and formative information: 
Complete and sufficient descriptions [difficult]
 
Description with timeframe, pace, and utilising the context
Until 10:30, Hohepa did no work because it was a weekend and they deserved a sleep in geez. They did a bit of steady painting until 12, at which point they retired briefly, possibly for a spot of tea or a sammich. At 1pm they went back to painting, and this time they went ALL OUT. For two more hours they painted, until 3pm. They paused to grab a biscuit and a glass of pineapple juice, before getting back to the job of painting their wall, whatever that was for anyway. They took their time that time, well, more than in the 1-3pm time frame anyway. Hohepa finished at 5pm.
 
Description with timeframe, pace, with minimal context 
9 a.m - 10.30 a.m: nothing, 10.30 a.m - 12 noon: a little working, 12 noon - 1 p.m: nothing, 1 p.m - 3 p.m: quite a lot, 3 p.m - 3.15 p.m: nothing, 3.15 p.m - 5 p.m: average amount of work.
 
Partial answers [easy]
 
Description with timeframe, minimal context, but no inclusion of pace
He wakes up at 10 and starts painting at 10:30 and has an hour rest at 12 to have lunch, he goes back at 1 and keeps going to 3, at 3 he has a 5 minute break for a snack and then keeps going to 5.
 
Description with timeframe, very little context (painting is implied), and no inclusion of pace
Started at about 10:30am, took a break at 12:00pm, started again at 1:00pm, small break at 3:00pm and finished at 5:00pm.
 
Description with timeframe, context, pace, and some detail missing (3-5pm)
His prep lasted till 10:30am. He had a 1 hour lunch break at 12pm. He worked extremely had from 1pm - 3pm.
 
Incomplete/Insufficient detail
Significant detail missing
Description has context and order, but no timeframe, and no inclusion of pace
Getting painting sorted, then painting, having a quick sleep, painting,then small break, then painting.
 
Summary statement
The wall had the most paint on it at 5:00pm and the least from 9:00am to about 10:30am.
Next steps: 
Complete and sufficient description with timeframe, pace and context
Students who gave a complete description could further explore the idea of gradient/slope by comparing the meaning of different slopes in the resources: Running stylesGetting to school
 
Description with timeframe, some context, and no inclusion of pace
Students who gave a description, but did not explain the significance of the slope could be asked what they notice is different about the three times that Hohepa painted (how the lines look different). Drawing horizontal lines across to the y-axis lines could illustrate that more work was done (noting that they represent 1.5, 2.0 , and 1.75 hours, respectively). Explore what it might mean if the lines are steeper. They could explore more simple graphs in the resources: Hemi's bike trip and Jonathan's journey to school. These resources use simpler scenarios with distance-time graphs. 
Using the graph in the resource Skyrocket and asking about the speed of the skyrocket might generate some interesting discussion around slope.
 
Description with timeframe, context, pace, but with some detail missing
Students who missed a smaller detail could be asked to carefully review the graph and their description and look for any information in the graph they have missed. They could follow the graph through and check their reading of the axes to ensure that they have a the correct change points. 
 
Start of a  description - but with a significant amount of detail missing
Students who gave an incomplete description could be asked what happened in the graph that has not been included in their description. They could annotate the graph for this task using what they have written and then the missing elements will become more obvious. Students could also explore simpler scenarios in the resources: Hemi's bike trip and Jonathan's journey to school. These graphs have descriptions already and could be used as models.
 
No description or detail derived from the graph
Students who could not begin to interpret or describe what was happening in the graph could be asked orally to identify a part of the graph and state what they think might be going on. Drawing their attention to the two axes and asking specific questions may cue them in to see the graph as representing a real situation, e.g., At 9 a.m. How much area was painted? followed by what Hohepa might be doing if there is nothing painted on the wall (sleeping, preparing, etc). 
 
Students may also need to first explore the idea of using a simpler graph to show what's going on. How hungry? is a resource that asks the students to draw a graph to show how hungry someone is over the course of a day. NOTE: this is a dot plot, and it does not explore the idea of slope, but it could be a starting point to see how things can be shown on graphs. Students could then look at simple line graphs in the resources: Hemi's bike trip and Jonathan's journey to school