YEAR 8 SRING TERM PROJECT ROOTS AND INDICES

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1 YEAR 8 SRING TERM PROJECT ROOTS AND INDICES Focus of the Project The aim of this The aim of this is to engage students in exploring ratio and/or probability. There is no expectation of teaching formal rules, methods or procedures at this stage. It is enough that students investigate and begin to appreciate the multiplicative nature of ratios. This will provide a basis on which to build when students meet ratio and proportion later in Year 8 and Probability in Year 9. The projects also offer the opportunity to consolidate work from earlier, such as basic numeracy and the chance to develop students problem solving skills. How to run this project Schools are free to run this project in whatever way fits their curriculum and allocation. For example, some schools have Maths days where some of the tasks could be used. Others may wish to use lesson time at the beginning or end of half terms to start some tasks and then let the students work through them as homeworks, either as individuals or in small groups. You might want to set aside a regular slot within other lessons to allow students to share their progress so far and discuss other options for continuing the project. How to mark this project As ever, we would never impose a system of schools, but this particular project would seem to lend itself very well to peer-assessment. In addition, students could be asked to work in pairs or groups to develop their own marking criteria. If the work is being done during lesson times, then progress can be checked and plenaries could be used to reinforce key findings; perhaps students could be asked to share something they have discovered during that lesson. Links to MM KS3 Programme of Study This project has direct links to: Year 7 Unit 1 Base 10 number system Year 8 Unit 1 Primes and factorisation Year 9 Unit 5 - Sequences Links to National Curriculum KS3 Programme of Study This project meets the requirements of the develop fluency, reason mathematically and solve problems elements of the new KS3 Curriculum. Several other elements of the Curriculum are directly met when using these projects: Use conventional notation for the priority of operations, including brackets, powers, roots and reciprocals. Use integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5 and distinguish between exact representations of roots and their decimal approximations. Round numbers to an appropriate degree of accuracy.

2 Suggested tasks for this Project Pascal s Triangle investigations Pascal s Triangle can be used to inspire a number of investigations linked to powers. Students could be asked to investigate various patterns that appear throughout the triangle. Adding up each row in the triangle produces the powers of 2: Powers of 11 can also be explored through the triangle, though this is a more challenging investigation = = = 121 etc

3 Once we get to the 6 th row, the first two-digit numbers first appear. The pattern for the powers of 11 does persist if multi-digit numerals are treated as having a single place value: s s 1 000s 100s 10s 1s So we have: = = 11 6 Square root Spiral Draw this right angled triangle in the middle of a piece of paper: Add the next right angled triangle keeping one side : Add on more triangles

4 Make a table and measure the lengths of the red lines (the hypotenuses of the triangles). Square each length. What do you notice? Why is this called a square root spiral? Triangle Length of hypotenuse Length squared x 1.0 = x 1.4 = x 1.7 = x 2.0 = 4.00 Chessboard investigations 1. Counting squares: This is a really nice investigation that generates the square numbers in two ways. Given a chessboard students are asked to find out how many different sizes of squares they can find on the board and how many of each type there are. It is worth printing out lots of copies of chess grids for students to draw and scribble on. For example, there are 64 1x1 squares and 49 2x2 squares. Students could also be asked to explore the area of these squares. There are 64 1x1 squares, each with an area of 1 square unit. There are 49 2x2 squares, each with an area of 4 square units...

5 2. Rice grain problem: This is a famous problem first written in the Shahnameh by Ferdowsi between 977 and 1010 CE. The story goes as follows: When the creator of the game of chess (in some tellings a mathematician named Sessa or Sissa) showed his invention to the ruler of the country, the ruler was so pleased that he gave the inventor the right to name his prize for the invention. The man, who was very clever, asked the king this: that for the first square of the chess board, he would receive one grain of rice), two for the second one, four on the third one, and so forth, doubling the amount each time. The ruler, arithmetically unaware, quickly accepted the inventor's offer, even getting offended by his perceived notion that the inventor was asking for such a low price, and ordered the treasurer to count and hand over the rice to the inventor. However, when the treasurer took more than a week to calculate the amount of rice, the ruler asked him for a reason for his tardiness. The treasurer then gave him the result of the calculation, and explained that it would take more than all the assets of the kingdom to give the inventor the reward. The story ends with the inventor becoming the new king. The idea here is to explore doubling numbers grows very quickly. This would make a really nice display piece and there is also the chance to use spreadsheets to generate results quickly, possibly to check answers. It is also possible to start with smaller chessboards, such as a 2x2 or 4x4 board and to estimate what would happen as the board got larger. Note: to enable students to appreciate a sense of the value of these numbers, on the entire board we end up with grains of rice (ask students to say this number in words!) this would weigh approximately tonnes (the possibility of comparisons here; how many elephants is this equal to?). This is a heap of larger than Mount Everest (again, lots of work can be done with the height and size of Everest) and 1000 times the global production of rice in 2010.

6 Exploring surds These tasks are mainly to allow students to explore using their calculators rather than developing an explicit understanding of the rules of surds or manipulation of surds. Task 1 Using the surds 2, 8, 10, 160, 320 and the operations and only, how many ways can you make 4? You can use each surd more than once. Record your results and write down anything you have noticed. Task 2 Lucy types 5 x 5 into her calculator. It gives the answer 5. She finds that 7 x 7 = 7. She conjectures that n x n = n. Is she right? Why? Task 3 Rajesh types 7 x 3 on his calculator and gets the answer 21. He conjectures that surds can be multiplied using this rule: n x m = nm. Is he right? How do you know? Does his rule work for the other three operations?