Trial & Improvement Activity What number to 1 d.p. must be multiplied to the number on your sticky note to reach 100? Take a sticky note Find your group Write your thinking process together

32 TRIALS ANSWER TOO BIG/TOO SMALL 3 3 x 32 = 96 TOO SMALL 4 4 x 32 = 128 TOO BIG 3.5 x 32 = 112 TOO BIG 3.5 3.2 x 32 = 102.4 TOO BIG 3.2 3.1 3.1 x 32 = 99.2 TOO SMALL 3.15 x 32 = 100.8 3.15 100.8 3.1 3.15 3.2 closer 3.1 to 1 d.p.

48 TRIALS ANSWER TOO BIG/TOO SMALL 2 .0 2 x 48 = 96 TOO SMALL 3 3 x 48 = 144 TOO BIG 2.5 2.5 x 48 = 120 TOO BIG 2.1 2.1 x 48 = 100.8 TOO BIG 2.05 2.05 x 48 = 98.4 98.4 2.0 2.05 2.1 2.1 TO 1 D.P.

11 TRIALS ANSWER TOO BIG/TOO SMALL 9 .0 9 x 11 = 99 TOO SMALL 10 10 x 11 = 110 TOO BIG 9.5 x 11 = 104.5 TOO BIG 9.5 9.1 x 11 = 100.1 TOO BIG 9.1 9.05 x 11 = 99.55 9.05 99.55 9.0 9.05 9.1 9.1 TO 1 D.P.

22 TRIALS ANSWER TOO BIG/TOO SMALL 4 4 x 22 = 88 TOO SMALL 5 5 x 22 = 110 TOO BIG 4.5 x 22 = 99 TOO SMALL 4.5 4.6 x 22 = 101.2 TOO BIG 4.6 4.55 4.55 x 22 = 100.1 100.1 4.5 4.55 4.6 4.5 TO 1 D.P.

45 TRIALS ANSWER TOO BIG/TOO SMALL 2 2 x 45 = 90 TOO SMALL 3 3 x 45 = 135 TOO BIG 2.5 x 45 = 112.5 TOO BIG 2.5 2.2 x 45 = 99 TOO SMALL 2.2 2.3 2.3 x 45 = 103.5 TOO BIG 2.25 x 45 = 101.25 2.25 101.25 2.2 2.25 2.3 2.2 TO 1 D.P.

Substitution Activity  

Let’s do it! Solve x2 + x = 53 correct to 1 decimal place First find the two whole number values of 𝑥 that will give the closest answers to 53: one will be a bit too small, the other will be a bit too big Then ZOOM in between these two numbers to find the two numbers with one decimal place that give the closest answers to 53: again one will be a bit too small, the other will be a bit too big To work out which one of those two numbers is the BEST answer you need to ZOOM in again to HALFWAY between them, then you will be able to see which one was closest to 53 – and sometimes you may be surprised This last step is the most commonly missed out part because sometimes it seems unnecessary. This means we need to find a number with one decimal place which gives the closest answer to 53 Let’s do it!

Solve x2 + x = 53 correct to 1 decimal place First find the two whole number values of 𝑥 that will give the closest answers to 53 x x2+x 6 42 too small 7 56 too big 6 7 10 42 56  

Solve x2 + x = 53 correct to 1 decimal place ZOOM in between these two numbers to find the two numbers with one decimal place that give the closest answers to 53 x x2+x 6 42 too small 7 56 too big 6 7 42 56 Let’s zoom in!

The best answer to 1dp will be either 6.7 or 6.8, but which one? Solve x2 + x = 53 correct to 1 decimal place Now we try sensible values to find the two best values of x with one decimal place. x x2+x 6 42 too small 7 56 too big 6.5 48.75 6.8 53.04 6.7 51.59 6 6.5 7 6.7 6.8 42 48.75 51.59 53.04 56 The best answer to 1dp will be either 6.7 or 6.8, but which one? Let’s try halfway for starters Let’s try a bit higher but still just one decimal place

Solve x2 + x = 53 correct to 1 decimal place ZOOM in between these two numbers to HALFWAY – then we’ll be able to see which side the true answer lies   x x2+x 6 42 too small 7 56 too big 6.5 48.75 6.8 53.04 6.7 51.59 6.75 52.3125 6.75 6.7 6.8 52.3125 51.59 53.04 We write: 𝑥 = 6.8 to 1dp Let’s zoom in! 53 is this side of half way It may seem like 6.8 was obviously the closest answer, but without this final checking stage you won’t know for sure

So In Summary… You write this: x x2+x 6 42 too small 7 56 too big 6.5 48.75 6.8 53.04 6.7 51.59 6.75 52.3125 Find the best two whole number solutions Zoom in to find the best two solutions to 1dp Zoom in halfway to find which one of them was the closest overall Answer: x = 6.8 to 1dp

Solve x2 - x = 17 correct to 1 decimal place First find the two whole number values of 𝑥 that will give the closest answers to 17 x x2 - x 5 20 too big 4 12 too small 5 4 10 20 12

The best answer to 1dp will be either 4.6 or 4.7, but which one? Solve x2 - x = 17 correct to 1 decimal place Now we try sensible values to find the two best values of x with one decimal place. x x2 - x 5 20 too big 4 12 too small 4.5 15.75 4.7 17.39 4.6 16.56 4 4.5 5 4.6 4.7 12 15.75 20 The best answer to 1dp will be either 4.6 or 4.7, but which one? Let’s try halfway for starters Let’s try a bit higher but still just one decimal place

Solve x2 - x = 17 correct to 1 decimal place ZOOM in between these two numbers to HALFWAY – then we’ll be able to see which side the true answer lies x x2 - x 5 20 too big 4 12 too small 4.5 15.75 4.7 17.39 4.6 16.56 4.65 16.97 4.65 4.6 4.7 16.56 17.39 16.97 Let’s zoom in! 17 is this side of half way We write: 𝑥 = 4.7 to 1d.p.

Classwork Worksheet

3 32 + 4 = 13 too big 2.5 2.52 + 4 = 10.25 too big 2.3 2.32 + 4 = 9.29 too big too small 2.2 2.22 + 4 = 8.84 2.25 2.252 + 4 = 9.0625 2.2 2.25 2.3 2.2 9.0625

2 23 + 2 = 10 too small 3 33 + 3 = 30 too big 2.5 too small 2.6 2.53 + 2.5 = 18.125 too small 2.6 2.63 + 2.6 = 20.176 too big 2.55 2.553 + 2.55 = 19.131375 2.5 2.55 2.6 2.5 19.13

3 32 + 4(3) = 21 too small 4 42 + 4(4) = 32 too big 3.5 too small 3.7 3.52 + 4(3.5) = 26.25 too small 3.7 3.72 + 4(3.7) = 28.49 too small 3.8 3.82 + 4(3.8) = 29.64 too big 3.75 3.752 + 4(3.75) = 29.0625 3.7 3.75 3.8 3.7 29.0625

6 too small 7 too big 6.5 too big 6.2 too small 6.3 too big 6.25 6.2 63 + 3(6) = 234 too small 7 73 + 3(7) = 364 too big 6.5 6.53 + 3(6.5) = 294.125 too big 6.2 6.23 + 3(6.2) = 256.928 too small 6.3 6.33 + 3(6.3) = 268.947 too big 6.25 6.253 + 3(6.25) = 262.890625 6.2 6.25 6.3 6.2 262.890625