Alien Estimates

This activity encourages students to use place value knowledge and additive part–whole strategies to estimate the results of calculations.
Students need to know numbers up to the hundreds of thousands to complete this activity. They need to be familiar with rounding numbers to the closest tidy number, for example, rounding 82 145 to the nearest tidy number in the thousands would give 82 000.
Some students may have problems with place value if, for example, they don’t know that for 82 000 + 45 000, they can calculate 82 + 45 in the thousands place. Once they realise they can calculate in this way, you will need to ensure that they don’t treat as ones units the numbers in the hundreds, thousands, ten thousands places, and so on. Place Value Houses are a useful way of clarifying any misconceptions that the students may have. (See The Power of Powers, pages 14–15, in Number Sense and Algebraic Thinking: Book One, Figure It Out, Level 3.)
This activity would be useful as an independent follow-up for a group that has just focused on rounding whole numbers to the closest tidy number in the tens, hundreds, or thousands.
With a guided teaching group, set the scene by asking about the students’ favourite computer or games-machine programmes and whether they’ve ever played a multi-person computer game.
Some students may be reluctant to estimate the answers in question 1 and may want to work out the calculation exactly. Encourage them to focus on what Rongomai has explained in her speech bubble rather than starting with the boxes of scores. Ask questions such as:
Where has Rongomai got the numbers 82 and 46 from? (The 82 is really 82 000, which is Ben’s score of 82 145 rounded down to the nearest tidy number in the thousands. The 46 refers to 46 000, which is Eseta’s score of 45 877 rounded up to the nearest tidy number in the thousands.)  Why do you think Rongomai decided to round the scores to the nearest thousand? (She wanted to make them into numbers she could add easily in her head, such as 82 000 + 46 000, but still be pretty accurate. If she’d rounded to the nearest tens of thousands, her estimation wouldn’t
have been as close to the actual total [80 000 + 50 000 = 130 000].)
What strategy might she have used to work out 82 + 46? How would you do it? (Rongomai could have used place value partitioning [80 + 40 = 120, 2 + 6 = 8, then 120 + 8 = 128] or added a tidy number and compensated [82 + 50 = 132, 132 – 4 = 128] or made equal adjustments [82 + 46 = 80 + 48 = 128].)
For question 2, before the students attempt to calculate the totals in round 2, ask: How would you round each person’s score if you were using Rongomai’s strategy?
For more estimating practice, the students could make up their own scores for round 3, follow the “deliberate mistake” step in question 2a ii, and swap with a classmate.
You could use questions 2b and 3 for formative assessment if the students are able to articulate the decisions they make when estimating.
Students who have difficulty rounding numbers to the nearest thousand may benefit from using an empty number line to help them visualise the distance of a number from its tidy number neighbours.
For a number like 69 034, tell the students that they’re trying to find out which tidy number in the thousands it is closest to. Start your number line at 60 000 and get the students to count by saying the number words with you as you mark 61 000, 62 000, 63 000, … up to 70 000 on it.
Then ask: Which two tidy numbers on the number line will the number 69 034 be between? (Circle 69 000 and 70 000.) These are its thousands tidy number neighbours. So which tidy number is 69 034 closest to on our number line?
Encourage the students to use imaging with questions such as:
Which thousands tidy numbers are going to be our number’s neighbours?
Which of these two tidy number neighbours is our number closest to?
The students using number properties may start talking in terms of the number of hundreds and how that will tell them that the number is closer to the next 1 000 up or to the 1 000 before.

Answers to Activity
 

1. Yes, Rongomai is correct. She rounded 82 145  down to 82 000 and 45 877 up to 46 000. Then she rounded 61 480 down to 61 000 and 67 952 up to 68 000. Ben had entered 1 instead of 7 on the calculator. The Martians’ round 1 score should
have been 129 432.
2. a. i. Round 2: Cyborgs 272 733, Martians 270 954
ii.–iii. Results will vary.
b. One possible strategy (Rongomai’s) is to round the numbers to the nearest 1 000
and work out what 69 000 plus 204 000 is for the Cyborgs and what 176 000 plus
95 000 is for the Martians team.
3. Estimates will vary. A sensible estimate is 300 000 + 140 000 = 440 000. Any estimate should focus on the digits in the high value places and ignore those in the lower value places.