Thinking Mathematically PROBLEMS AND STRATEGIES TO ENCOURAGE MATHEMATICAL

Thinking Mathematically PROBLEMS AND STRATEGIES TO ENCOURAGE MATHEMATICAL THINKING IN THE CLASSROOM Jonathan Hall Maths Lead Practitioner Leeds City Academy @Study. Maths. Bot. com

Workshop Aims: • TRY A VARIETY OF PROBLEMS TO ENCOURAGE MATHEMATICAL THINKING IN THE CLASSROOM. • USE THE RUBRIC ‘ENTRY, ATTACK, REVIEW’ WHEN TACKLING A PROBLEM; STARTING WITH SPECIALISED CASES BEFORE WORKING TOWARDS GENERALISED SOLUTIONS. • ENCOURAGE USE OF RIGOROUS MATHEMATICAL LANGUAGE.

• CHOOSE ANY 3 DIGITS (AT LEAST 1 DIFFERENT FROM THE REST). • WRITE DOWN THE LARGEST 3 -DIGIT NUMBER YOU CAN MAKE. • WRITE DOWN THE SMALLEST 3 -DIGIT NUMBER YOU CAN MAKE. • FIND THE DIFFERENCE BETWEEN THEM. • REVERSE THE DIGITS OF YOUR ANSWER AND ADD THEM TOGETHER. 1089 Why does this work?

Thinking Mathematically J. Mason, L. Burton, K. Stacey

Geoboards • Make a rectangle, triangle, parallelogram and trapezium. • The shapes must not be congruent to each other. • The shapes must each have an area of 16 square units. • The shapes must not touch each other. mathsbot. com/manipulatives/geoboard

Geoboards mathsbot. com/manipulatives/geoboard

Geoboards Make 4 rectangles, each with an area of 12 square units. The rectangles must not touch each other. The rectangles must not be congruent to each other. Can you do it for an area of 15 square units? Why is it harder? mathsbot. com/manipulatives/geoboard

Geoboards How many different triangles with an area of 6 square units can you make on a 10 x 10 geoboard? Make a rectangle, triangle, parallelogram and trapezium all with an area of 16 square units. mathsbot. com/manipulatives/geoboard

Chessboard Squares It was once claimed there are 204 squares on a chessboard. Can you justify this claim?

Chessboard Squares Square Size 8 x 8 7 x 7 6 x 6 5 x 5 4 x 4 3 x 3 2 x 2 1 x 1 Total Square Size 8 x 8 7 x 7 6 x 6 5 x 5 4 x 4 3 x 3 2 x 2 1 x 1 • What about an n x n board? • What about rectangles? • What if we extend to work in 3 dimensions. Total 1 4 9 16 25 36 49 64 204

Nearest Squares

Nearest Squares a² b² c² b² - a² c² - b² Diff 1 4 9 3 5 2 4 9 16 5 7 2 9 16 25 7 9 2 16 25 36 9 11 2 25 36 49 11 13 2 2704 2809 2916 105 107 2 mathsbot. com/manipulatives/tiles

Tethered Goat A goat is tied by a 6 metre rope to the outside corner of a rectangular shed. The shed measures 4 metres by 5 metres and is in the middle of a large grassy field. What area of grass can the goat eat?

Tethered Goat

Palindromes “All four-digit palindromes are divisible by 11” Is this true?

Palindromes 1001 1111 1221 1331 1441 1551 1661 1771 1881 1991 +110 +110 +110 1991 2002 2112 2222 2332 +110 +110

Paper Strip A long strip of paper is folded in half so the two ends meet. Now repeat the process with the new strip. How many creases are there once the paper is unfolded? How many creases will there be if the operation is repeated 10 times?

Paper Strip What if you folded the paper into thirds each time? Quarters? N Parts? Folds 11 22 33 44 55 66 77 88 99 10 10 nn Creases 11 1 33 3 77 7 15 15 15 … 31 63 127 255 511 1023 2^n - 1 Parts 2 2 4 4 8 8 1616 32 64 128 256 512 1024 2^n

What is the length of the side of the square? How many ways can you solve the problem? Trigonometry, similar shapes, area, Pythagoras? What do you notice about the side lengths and your solution? What questions could you ask?

What is the length of the side of the square? Notice: 3 × 4 = 12 and 3 + 4 = 7? What questions do you ask yourself? Can you predict, conjecture, generalise?

One Sum Choose two fractions which sum to 1. a) Square the larger and add it to the smaller. b) Square the smaller and add it to the larger. Which answer is the biggest?

Specialising and Generalising

Thinking Mathematically J. Mason, L. Burton, K. Stacey

J. Mason, L. Burton, K. Stacey EEF RAG