These are level 3 number, geometry and statistics problems from the Figure It Out series.
A PDF of the student activity is included.
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find fractions of regions (Problem 1)
solve addition problems involving money (Problem 2)
explore rotational symmetry (Problem 3)
investigate bar and pie graphs (Problem 4)
Problem One
Encourage students to visualise and use reasoning before they measure the rectangles, though this measuring can be used to confirm their thinking. They will have to use fractional knowledge in this problem.
This could be confirmed by fitting the small rectangle into the large one 16 times, as shown.
Using similar reasoning, a quarter of the small rectangle must be 1/4 of 1/16, which is 1/64 (one sixty fourth).
Problem Two
Sarah will receive $2 – $1.20 = 80c in change. Students will need to look for combinations of four coins that add to 80 cents. The results can be organised into a table to eliminate combinations that don’t work and to avoid duplication.
Systematically working through by starting with combinations involving 5 cent coins will find all the possibilities.
Problem Three
This problem uses rotational symmetry. All the numbers used in the expression are either:
• images of themselves after a half turn, for example:
or
• mirror images of another number.
The matching numbers are shown below.
The answers are the same because the same numbers are added.
Problem Four
One way to confirm that these graphs are showing the same data is to imagine how the bar graph could be transformed into a pie graph.
Another method is to apply fractional knowledge to the numbers involved. The bars have heights of 4, 6, and 2 respectively, so there are 4 + 6 + 2 = 12 items of data. The red bar has height 4, so it is 4/12 or 1/3 of the number of the data items. The blue has height 6, so it is 6/12 or 1/2 of the data items. The pie chart shows fractions of 1/3, 1/2, and 1/6, so the representations match.
Answers to Problems
1. a. 1/16 (0.0625)
b. 1/64
2. 4 x 20c; 50c, 10c, 10c, and 10c; or 50c, 20c, 5c, and 5c
3. a. 3 481
b. You get the same answer.
c. When the book is turned upside down, you are looking at exactly the same set of
numbers being added together as before because 8 and 1 have half-turn symmetry, 9
is a half-turn image of 6, and 2 is a half-turn image of 5.
4. Yes. The first bar is showing 4 out of 12 units, the second is showing 6 out of 12, and the third 2 out of 12. Expressed as fractions, this is 4/12 = 2/6, 6/12 = 3/6, and 2/12 = 1/6. These are the same fractions shown by the pie graph.
Students’ stories will vary.