Solve problems involving ratios.
Number Framework Stage 8
Discuss a situation where the students have encountered percentages in their daily
life. They will often suggest sports (for example, shooting for goal), shopping (such
as discounts or GST), and, in country areas, calving or lambing percentages. Tell the
students that the % sign comes from the “out of” symbol, /, and the two zeros from
100. It means, “out of 100”.
Problem: “In a game of netball, Irene gets in 43 out of her 50 shots. Sharelle takes 20
shots and gets in 17. Who is the better shot?”
Tell the students that percentages are used to compare fractions. In Irene’s case, the
fraction is 43/50 . Doubling 43 calculates the shooting percentage because 43/50 is equivalent
to 86/100 (86 out of 100). Represent this on a double number line.
Ask the students to work out what Sharelle’s shooting percentage was for the same
game. Represent this on a double number line to show that fi nding a percentage is
like mapping a proportion onto a base of 100.
Pose the students a percentage problem that can be modelled with the percentage strips.
For example: “Tony got in 18 out of his 24 shots. What percentage did he shoot?”
Mapping 18 out of 24 onto a base of 100 gives 75%.
Pose similar problems that the students can solve by aligning differently based
strips with the 100-base strip. Examples might be:
16 out of 32 (50%) 9 out of 36 (25%) 10 out of 25 (40%)
12 out of 16 (75%) 12 out of 40 (30%) 4 out of 20 (20%)
Show the students the base strip, but have the percentage strip aligned to it and
turned over so they can’t see the beads. Give the students “out of” problems and
have them estimate the percentage by visualising.
For example, pose six out of 16. Mark six with a paper clip. The students should
estimate the percentage as just below 40% or greater than 33.3% (one-third). A
calculator can be used to work out the exact percentage by keying in 6 ÷ 16%. The
percentage strip can then be turned over to check the estimate. Ask how else they
could have estimated the percentage if there had been no strips.
Look for ideas like “Six out of 16 is the same as three out of eight, and that is half of
three out of four” or “There are over six sixteens in 100. Six times six is 36, so it will
be more than 36 percent.”
Pose similar imaging problems like:
8 out of 20 (40%) 15 out of 25 (60%) 4 out of 16 (25%)
32 out of 40 (80%) 20 out of 32 (62.5%) 14 out of 36 (39%)
Focus on strategies based on the numbers involved that could have been used to
estimate the percentages.
The students can play the game of Percents (see Material Master 7–5) to consolidate
Using Number Properties
Give the students percentage problems to solve. Pose these problems in contexts
of sports scores, shopping discounts or mark-ups, or lambing percentages. Pose
some problems where duplication of the base onto 100 is not easy. For example, 25
is easily mapped onto 100 through multiplying by four, whereas 40 is not so easily
mapped (although students should be encouraged to recognise that 2.5 _ 40 = 100).
Examples that involve percentages greater than 100 should also be used.
Some examples might be:
18/24 = ? %? (75%)
25/40 = ? %? (62.5%)
18/27 = ? %? (66.6%)
8/32 = ? %? (25%)
24/16 = ? %? (150%)
55/20 = ? %? (275%)
Get the students to record their thinking using double number lines or ratio tables,
e.g., 27/36 = 75%.
Use brochures from local retailers. Tell the students that one shop has a “25% off”
sale, another has a “40% off” sale, and a third has a “one-third off” sale. Give the
students an arbitrary budget to spend at the three shops.