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This is a level 4 number activity from the Figure It Out series. It relates to Stage 7 of the Number Framework.
find fractions of whole numbers
Number Framework Links
Use this activity to help students consolidate and apply their knowledge of fractions (stages 6 and 7).
Question 1 has 12 parts to it, so your students may find it helpful if they create a suitable table and then complete it as they work out each fraction:
Students should find it easy to come up with numerous answers for question 2 without needing any system. You may, however, like to suggest an approach that will help them find many solutions at the same time as they create meaning for algebraic representation and substitution.
Suggest using this shorthand: A (for one 6-star packet), B (8 stars), C (12 stars), and D (20 stars).
We can see that 144 is divisible by 6 (24A packets make 144), and 8 (18B packets make 144), and 12 (12C packets make 144). Also, B + C = D and 2A = C, so by substitution, we can get many more solutions:
6D + 2C = 144
5D + 3C + B = 144 (replacing 1D with one B + C)
4D + 4C + 2B = 144 (replacing 1D with one B + C)
3D + 5C + 3B = 144 (replacing 1D with one B + C)
and so on until we end up with 8C + 6B = 144.
We know that 12C = 144 and 2A = C, so we can keep replacing 1C with 2A to get another group of solutions:
2A + 11C = 144 (replacing 1C with 2A)
4A + 10C = 144 (replacing 1C with 2A)
6A + 9C = 144 (replacing 1C with 2A)
and so on until we end up with 24A = 144.
Other combinations such as 6A + 6B + 3D can provide the starting point for further lists of possibilities.
Question 3 asks students to calculate 1/3 of 144 and 1/4 of 144. They should share their strategies for these calculations. Those who know their basic facts are at stage 7 and more likely to use the short form of division.
Question 4 requires students to express in its simplest form. Although these numbers lie outside the range of known facts for many students, they have been working with 144 in questions 2 and 3 and should have little trouble finding a strategy that they can use to simplify the fraction. Those who remember that 144 = 12 x 12 have access to the most efficient strategy of all, the one that avoids finding the product 12 x 9 and sees that Anita is in effect using just 9 stars out of every 12, so she uses 9/12 = 3/4 of the total.
Answers to Activity
1. 3-star clusters:
1/2 of a 6-pack; 3/8 of an 8-pack; 1/4 of a 12-pack; 3/20 of a 20-pack.
4/6 or 2/3 of a 6-pack; 1/2 of an 8-pack; 1/3 of a 12-pack; 1/5 of a 20-pack.
5/6 of a 6-pack; 5/8 of an 8-pack; 5/12 of a 12-pack; 1/4 of a 20-pack.
2. Many answers are possible. They include: twenty four 6-packs; twelve 12-packs; six 6-packs + six 8-packs + three 20-packs.
3. 16 x 3-star clusters. (1/3 of 144 = 48)
9 x 4-star clusters. (1/4 of 144 = 36)
12 x 5-star clusters. (144 – 48 – 36 = 60)
4. . (12 x 9 = 108, and 108 is 3/4 of 144.)