Pene's puzzles
This is a level 3 algebra strand activity from the Figure It Out series. A PDF of the student activity is included.
About this resource
Figure It Out is a series of 80 books published between 1999 and 2009 to support teaching and learning in New Zealand classrooms.
This resource provides the teachers’ notes and answers for one activity from the Figure It Out series. A printable PDF of the student activity can be downloaded from the materials that come with this resource.
Specific learning outcomes:
- Solve sets of equations.
Pene's puzzles
Achievement objectives
NA3-6: Record and interpret additive and simple multiplicative strategies, using words, diagrams, and symbols, with an understanding of equality.
Required materials
- Figure It Out, Level 3, Algebra, "Pene's puzzles", pages 22–23
- calculator
- classmate
See Materials that come with this resource to download:
- Pene's puzzles activity (.pdf)
Activity
The sequence in these patterns leads students towards developing rules. Students may wish to complete the pattern through to the tenth element in order to answer question 2, but do encourage them to also use more efficient methods. See answers and teachers’ notes: Levels 2–3 Algebra, Figure It Out, page 25.
If students aren’t convinced that Mark’s answer for question 1 is always three, have them try the puzzle several times with different numbers. You may like to work through the puzzle with students to help them identify what is happening. Using the notation in Mark’s speech bubble is especially useful because it makes students more familiar with algebraic notation, and writing the puzzle down is much easier than doing it mentally, as Mark has done.
In order to always get an answer of three, the steps in the equation must, to a certain extent, cancel each other out. So students need to look for inverse operations that do cancel each other out:
The operations x 2 and ÷ 2 cancel each other out. Six has been added on, but this is also divided by two, so it is now three. The original number is cancelled out by subtraction, leaving an answer of three.
Remind students to keep these points in mind when they make up their own puzzles. Most of the operations that they include in their puzzles will have to cancel each other out.
You may find the following algebraic proofs interesting (although students will not yet be using this level of algebra).
1.
Let n be the number thought of.
Take away the first number thought of: n + 3 – n = 3.
2.
Let n be the number thought of.
Double it: 2n.
Take away 7: 2n – 7.
Add 21: 2n – 7 + 21 = 2n + 14.
Divide by 2: n + 7.
Subtract n: n + 7 – n = 7.
Activity 1
1.
a. 5 ÷ 5 = 1
b. 5 x 5 = 25
c. 6 x 1 + 1 = 7
2.
a. 10 ÷ 10 = 1
b. 10 x 10 = 100
c. 11 x 1 + 1 = 12
3.
Answers will vary.
Activity 2
1.
a. Yes
b. One explanation is: Mark knew that x 2 and ÷ 2 cancelled each other out, 6 ÷ 2 = 3, ? and ? = 0. The only number left is 3, which is the answer.
2.
a. Yes
b. This can be explained in 2 steps.
Step 1:
- The puzzle can be written as (? x 2 – 7 + 21) ÷ 2 = ?
- This can be simplified to (? x 2 + 14) ÷ 2 = ? and then to ? + 7 = ?
Step 2:
- ? – ? = 7. (? will always be your number plus 7. When you subtract your number, you will always be left with 7 as an answer.)
3.
Puzzles and solutions will vary.
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