Hello and welcome to the Math Kangaroo Media Library. You are about to look at interactive solutions to levels 1 and 2 of the 2011 competition. You will notice that perhaps some of the solutions presented here are slightly different from the solutions provided and the suggested solutions, and possibly you have your own ideas on how these problems should be solved. So please, as you follow along, understand these differences as that will help you prepare for your next Math Kangaroo competition. If you have any questions or comments, feel free to contact me at the email address provided. My name is Luke and I am a past Math Kangaroo participant. I hope that you will find this presentation useful in preparing for your next Math Kangaroo competition. Question number 1. Consecutive positive numbers were placed in the cells of the table below. Here we have a table with some numbers and a question mark for a missing number. What number is missing from the middle cell? So our job is to discover the pattern in this table and decide what number should replace the question mark. We have a clue that the entries of the table are made up of consecutive positive numbers, meaning that we cannot leave gaps between our numbers. So there is just one number here between 2 and 4 that fits and no other number can take the spot. Maybe you already guessed that the answer is 3, but let's go through why it has to be 3. If we were to write down the number 1, the first number in our table, we could also tally one item and that would be one tally mark. If we wrote 2 as a number, that is one more tally mark than previously. So now I have 2 marks. And if I continue to 3, I need to add one more tally mark to the 2 I had before. And I can keep going like that until I have all the numbers in my table. So with 4, I add one more tally mark to the 3 I had before. And with 5, we have the 4 tally marks from before and then one more. And so we see that we have the 1 in our table, we have the 2, we have the 4, and we have the 5. And there is no number corresponding to a situation where we have anywhere in between 4 and 2 tally marks unless it is the number 3. So that is the missing number and our answer here is C. Question number 2, 6 plus 2 is equal to which of the following numbers? Let's rewrite our equation again and then use tallies to find out the answer. 6 can be represented by 1, 2, 3, 4, 5, and 6 tally marks. To that, we're going to add 2 which is represented by 1, 2 tally marks. And as a final answer, we will write down all the tally marks all at once. So we have our 2, 3, 4, 5, 6 from before, and then 2 more, 1, 2, that makes a total of 8 marks. So the answer is 8 and now we know that 6 plus 2 is equal to 8 and we choose the answer D. Question number 3, Sharon had 10 dolls. She gave Betty one of her dolls. How many dolls does Sharon have now? Let's draw some pictures to keep track of the dolls Sharon had. So on the left here, I will have 10 pictures of dolls. Here is the first one, 2, 3, 1 more makes 4, now we have 5, and we can double that, 6, 7, 1 more makes 8, 9, and finally 10. We can even label these to practice our counting. We have 1, 2, 3, 4, 5, 6, 7, 8, 9, and finally 10. But we know that one of these dolls was given to Betty. So let's take one and move it over from Sharon's side to Betty's side and we see now that Betty has 1 doll and Sharon no longer has 10, she is left with 9. So after giving 1 doll away to Betty, Sharon's 10 dolls became 9 and that's how many she has left. So the answer here is D. Question number 4. There are 2 boys and 2 dogs and nobody else on the playground. How many legs are there on this playground? Let's draw pictures of all the dogs and all the boys present and beginning with the dogs, we need to have exactly 2 of them on the playground and then moving on to the boys, we will need to have exactly 2 of them as well. So here are pictures of all the dogs and all the boys and nobody else. Now we can count legs. Each dog has 1, 2, 3, 4 legs and so we do that again and then each boy has 1, 2, 2 legs. And our job is to find out how many legs there are on the playground in total. So we add 4 plus 4 plus 2 plus 2 and we can keep track of the tallies to obtain our answer. We have 1, 2, 3, 4, 5, 6, 7, 8 tallies, 9, 10, 11 and 12 accounting for all the legs we counted between all the dogs and then all the legs we counted between all the boys. So the answer comes out to 12 and that is A, the answer to question number 4. Question number 5. Which month sometimes has only 29 days? We know that February has 28 days and usually months have 30 or 31 days. We also know that usually years have 365 days but every once in a while a leap year will have 366 days. This extra day is added on to February, the shortest month, to give it a total of 29 days. So the next time we will have a February with 29 days will be in the leap year 2016. The answer to question number 5 is B, February. Question number 6. Seven students and a teacher are ready for a snack. There are seven glasses of milk right here, eight candy bars and a cup of coffee ready for them. Each student will have the same snack. How many candy bars will the teacher get with his coffee? We can imagine that as everybody comes up to the table, the students each grab a candy bar and then a glass of milk. So we can imagine that each candy bar belongs to a glass of milk. And so we draw arrows from the candy bars to the glasses of milk because they will be taken away two at a time like this until each student has his or her snack. And so we continue doing this until we run out of glasses of milk. Now the teacher is also having a snack and so the last candy bar here belongs with the cup of coffee. That's the teacher's snack. And so we see that the last candy bar belongs to the teacher and in fact everybody has a drink and candy bar, including the teacher. So he has one candy bar with his coffee. Question number seven. What is the sum of the digits in the number 2011? First, let's write down the number 2011 again, 2011, and then let's decide where the digits are. And we know that digits, we can remind ourselves, are the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. There are 10 digits. And so the digits here in 2011 are 2, 0, 1, and 1. These are the digits. Now we have to add them up. So we add the first digit to the second digit, which is 0, add 0 to the third digit, which is 1, and finally to the whole sum here, add the last digit and count them up. We can use tallies to help us out. 2 is 2 tallies, 0 is no tallies, and then 1 gets a tally each. So together we have 1, 2, 3, 4 tallies, and the answer should be the number 4. In fact, 2 plus 0 is 2, 1 plus 1 is 2, and so 2 plus 2 will make 4. And that is the sum of all the digits present in the number 2011. Answer C. Question number 8. Katie's doll is wearing a dress, has two braids, and is holding one flower in her hand. Which picture shows Katie's doll? Let's look at each of the pictures carefully and see if we can find a doll wearing a dress, holding one flower, and having two braids. We see that in picture A, the doll is not wearing a dress. She is holding a flower, but she does not have braids, so this cannot be Katie's doll. Now this is the only doll that is not wearing a dress, but not the only doll not having any braids. In picture C, we have a doll that also does not have any braids, so she also cannot be Katie's doll. Now each doll is holding flowers, but Katie's doll has only one flower, and the doll in picture D has two, so she also cannot be Katie's doll. And that leaves us with the picture B. We have a doll wearing a dress. She has two braids and one flower, so this must be Katie's doll. And we choose answer B.

Math Kangaroo

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