Purpose
This problem solving activity has a number focus.
Specific Learning Outcomes
- Multiply large numbers by 8.
- Find and describe patterns in numbers.
Description of Mathematics
In this problem students are challenged to find a number pattern. They may be able to generalise the pattern more readily if they first work without a calculator, before using a calculator to confirm their thinking.
Whilst much of mathematics is about finding patterns, the greater challenge, as in this problem, is to see how long they continue.
Activity
The Problem
We think that you’ll find a pattern in the numbers below.
8 x 8 + 13 =
88 x 8 + 13 =
888 x 8 + 13 =
8888 x 8 + 13 =
88888 x 8 + 13 =
Does the pattern extend indefinitely?
Teaching Sequence
- Pose the students the problem and have them complete a couple of calculations.
- Have them continue the calculations in their groups as they look for a pattern.
- Help groups as required though this is a pattern that most groups should be able to see.
- Pose the extension problem to those who are ready.
- Have a few groups report on what they have done. Make sure that everyone understands the pattern. Take care over the explanation.
- Give all students time to write up their solution. This should consolidate their understanding of what is happening in the pattern.
Extension
Have a look for the pattern here.
1 x 1 =
11 x 11 =
111 x 111 =
1111 x 1111 =
11111 x 11111 =
Does the pattern extend indefinitely?
Solution
The answers are 77, 717, 7117, 71117 and 711117.
So the conjecture (guess) is that 8…8 x 8 + 13 = 71…17. But we need to be a little more accurate than that. How many 8s will give us how many 1s? Well one 8 gives us no 1s, two 8s gives us one 1 and so on. So it looks as though if we have six 8s we’ll have five 1s, if we have ten 8s we’ll have nine 1s and we have n 8s we’ll have n - 1 1s. In other words we’ll get one less 1 than we have 8s. But how can we show this?
Let’s forget about the 13 for a minute. Then 8 x 8 = 64, 88 x 8 = 704, 888 x 8 = 7104, 8888 x 8 = 71104 and so on. Each time we add another 8 we add 64 to the first digit of the previous answer. The result is to change 71…104 with say m – 2 1s (coming from m 8s) into 71…104 with m – 1 1s. So one new 1 is added at each step. Now adding 13 at each stage we change the 04 into 17, to give another 1. So if m 8s give m – 2 1s in the 71…104 they give m – 1 1s in the 71…17. This is what we had guessed.
Solution to the Extension
Here the answers are 1, 121, 12321, 1234321 and 123454321. This pattern continues up to 12345678987654321 when carry-overs start to occur and mess up the pattern.
Little halves and Big Quarters
Find equivalent fractions and order fractions.
Order fractions, decimals and percentages.
Number Framework Stages 7 and 8
Ask the students whether 1/2 > 1/4 or 1/4 < 1/2 . The expected answer is 1/2 > 1/4 . Now pose this problem:
“Annabelle earns $100 a week, and Maureen earns $400 a week.” Discuss why, in this case, one quarter of Maureen’s weekly pay would be bigger than half of Annabelle’s weekly pay! Record on the board or modelling book, “Half of a small amount can be smaller than a quarter of a larger amount”.
Discuss why, when we are making comparisons between fractions, we need to be careful about what the respective “wholes” are.
Extension Activity
Ask the students to work out in this table whether or not half of Annabelle’s pay is the same as a quarter of Maureen’s pay.
Discuss the general rule: Half of Annabelle’s pay is the same as a quarter of Maureen’s pay provided Maureen earns exactly twice as much as Annabelle.
Pose similar problems like: “One half of Peter’s pay is the same as one-third of Nicola’s pay.”
“When can this statement be true?” “Come up with some numbers that work.” “Try to find the general rule for this statement.” (They are equal provided Peter earns exactly two-thirds of Nicola’s pay.)
Factor Cards
This activity provides students with a fun, game context in which to practice their factorisation skills. This activity builds the knowledge to help develop multiplicative strategies.
use their knowledge of basic facts to identify common factors of numbers to 100.
Eights
This problem solving activity has a number focus.
8 x 8 + 13 =
88 x 8 + 13 =
888 x 8 + 13 =
8888 x 8 + 13 =
88888 x 8 + 13 =
Does the pattern extend indefinitely?
In this problem students are challenged to find a number pattern. They may be able to generalise the pattern more readily if they first work without a calculator, before using a calculator to confirm their thinking.
Whilst much of mathematics is about finding patterns, the greater challenge, as in this problem, is to see how long they continue.
The Problem
We think that you’ll find a pattern in the numbers below.
8 x 8 + 13 =
88 x 8 + 13 =
888 x 8 + 13 =
8888 x 8 + 13 =
88888 x 8 + 13 =
Does the pattern extend indefinitely?
Teaching Sequence
Extension
Have a look for the pattern here.
1 x 1 =
11 x 11 =
111 x 111 =
1111 x 1111 =
11111 x 11111 =
Does the pattern extend indefinitely?
Solution
The answers are 77, 717, 7117, 71117 and 711117.
So the conjecture (guess) is that 8…8 x 8 + 13 = 71…17. But we need to be a little more accurate than that. How many 8s will give us how many 1s? Well one 8 gives us no 1s, two 8s gives us one 1 and so on. So it looks as though if we have six 8s we’ll have five 1s, if we have ten 8s we’ll have nine 1s and we have n 8s we’ll have n - 1 1s. In other words we’ll get one less 1 than we have 8s. But how can we show this?
Let’s forget about the 13 for a minute. Then 8 x 8 = 64, 88 x 8 = 704, 888 x 8 = 7104, 8888 x 8 = 71104 and so on. Each time we add another 8 we add 64 to the first digit of the previous answer. The result is to change 71…104 with say m – 2 1s (coming from m 8s) into 71…104 with m – 1 1s. So one new 1 is added at each step. Now adding 13 at each stage we change the 04 into 17, to give another 1. So if m 8s give m – 2 1s in the 71…104 they give m – 1 1s in the 71…17. This is what we had guessed.
Solution to the Extension
Here the answers are 1, 121, 12321, 1234321 and 123454321. This pattern continues up to 12345678987654321 when carry-overs start to occur and mess up the pattern.
Tennis and golf players
This problem solving activity has a number focus.
In fact 80% play tennis and 70% play golf.
What percentage play both games?
This problem involves reasoning with percentages. Percentages are fractions that are used to express the whole divided into one hundred equal parts. One percent means one-hundredth part of a whole or unit (e.g. 1/100), and the notation used is 1%. In the extension to this problem the students need to find the lowest common multiples of the numbers 20, 30 and 50. Your students should have some knowledge of percentages to successfully work with this problem.
One approach to this problem is to draw a diagram, representing the problem with intersecting sets. As Venn diagrams are sometimes difficult for students to understand it can be helpful to build them one step at a time, with several examples being explored at each step.
The Problem
In a class in the school down the road, everyone plays tennis or golf or both. In fact 80% play tennis and 70% play golf. What percentage play both games?
Teaching Sequence
What does 80% of the class mean?
What does 70% of the class mean?
How do you know what percentage play both tennis and golf?
How do you know what percentage of the class do not play each sport?
How has your diagram helped you to solve the problem?
What information did you need to include in your diagram?
Extension
What is the smallest number of students in that class?
Solution
If 80% of the class plays tennis, 20% don’t play tennis. This 20% must be part of the golf players (they play golf only). Similarly, if 70% of the class plays golf, 30% don’t play golf. This 30% must be part of the tennis players (they play tennis only). This means that the remaining 50% play both tennis and golf.
Solution to the Extension
Find the smallest number so that the various percentages of the Venn Diagram are whole numbers. 20%, 30% and 50% of the class need to be whole numbers. This will be the case if 20, 30 and 50 are in each group. It is also the case with 10, 15 and 25. So how do we get the smallest class?
What is the biggest number that goes into 20, 30 and 50. Once we’ve found that, if we divide 20, 30 and 50 by that amount we will have the smallest numbers we want. This biggest divisor is 10. So the separate amounts are 2, 3 and 5. The class has only 10 students.
Free Cell
This problem solving activity has a number focus.
I’m addicted to Free Cell, a game on our computer.
At the moment I’ve won 3300 games, which is 60% of the games I've played.
I’d like to be able to say that I had won two-thirds of the games that I had played.
How many games would I have to win in a row to get to the two-thirds winning mark?
To solve this problem the students must read carefully the information they are given, and decide how to use this to find an unknown amount.
An algebraic approach is desirable, but it may not be intuitively used by students. To successfully solve this problem, students need to have had experience with finding fractions and percentages of whole numbers.
The algebraic solution given will provide a foundation for a valuable discussion with students.
The Problem
I’m addicted to Free Cell, a game on our computer. At the moment I’ve won 3300 games, which is 60% of the games I've played. I’d like to be able to say that I had won two-thirds of the games that I had played. How many games would I have to win in a row to get to the two-thirds winning mark?
Teaching Sequence
What is a percentage?
What is a fraction?
How can we convert fractions like 1/3 and 2/3 to a percentage?
Would you rather have 2/3 or 60% of a chocolate bar? Why?
Show me how you calculate 2/3 of 120 using the calculator.
Tell me how you started the problem. Why did you start in that way?
How can you find 60% of a whole number? How can you apply that process to this problem?
Are you convinced that you have the correct answer? Why?
Solution
The information given is:
I have won 60% of my games. This number is 3300 so I can find the number of games that I have played. This is because 3300/games played = 60/100. So games played = (3300 x 100)/60 = 5500.
I’m going to play some more games and I’m going to win them all. So 2/3 = (3300 + more games) / (5500 + more games). In the table below, I’ll let the fraction on the right be F. This is now set up for a ‘guess and improve’ strategy and the use of a table.
So if I play another 1100 games and win them all, then I shall have won two-thirds of all the games I have played.
Algebraic Approach
We already know that 2/3 = (3300 + more games) / (5500 + more games). So let m = more games and we then have 2/3 = (330 + m)/ (5500 + m). Multiplying both sides by 3 and 5500 + m gives 2(5500 + m) = 3(3300 + m),
so 11000 + 2m = 9900 + 3m,
or m = 1100.
Who Gets More?
Find equivalent fractions and order fractions.
Order fractions, decimals and percentages.
Number Framework Stages 7 and 8
Ordinary squared paper
Set a comparison problem: “Twins receive identical cakes for their birthday. Jose cuts her cake into fifths and Michael cuts his into quarters. Jose eats of 4/5 her cake and Michael eats 3/4. Who eats more cake?”
Discuss why a 5 by 4 cake allows fifths and quarters to be drawn easily. The students draw two 5 by 4 grids on the large blank squared paper. The students shade the fractions.
Discuss why adding vertical lines and horizontal lines respectively is a good idea; it creates twentieths on both cakes.
Discuss why 4/5 > 3/4 follows from the pictures. (The pictures show 16/20 > 15/20 so 4/5 > 3/4 .
Compare this pair of fractions on the large squared paper: 2/3 and 3/4.
Repeat with these pairs of fractions on the ordinary squared paper: Compare 5/6 and 4/5, 7/10 and 2/3, 5/8 and 3/5 ...
Repeat with these pairs of fractions without any drawings: 2/3 and 13/20, 7/11 and 3/5, 69/100 and
Extension Activity
Order these sets of three fractions from smallest to largest each time: 3/5, 2/3, and 1/2; 7/10, 2/3, and 1/2,...
Fraction Benchmarks
In this unit students develop important reference points or benchmarks for zero, one half and one. They use these benchmarks to help compare the relative sizes of fractions, through estimating, ordering and placing fractions on a number line.
This unit builds on the following key conceptual understandings about fractional parts.
An understanding of fractional parts supports students to develop a sense for the size of fractions. This unit helps students develop an intuitive feel for zero, one-half and one, as useful benchmarks for ordering fractions. Understanding why a fraction is close to zero, one half or one helps students develop a number sense for fractions.
Students can be supported through the learning opportunities in this unit by differentiating the complexity of the tasks, and by adapting the contexts. Ways to support students include:
The contexts for this unit are purely mathematical but can be adapted to suit the interests and cultural backgrounds of your students. Fractions can be applied to a wide variety of problem contexts, including making and sharing food, constructing items, travelling distances, working with money, and sharing earnings. The concept of equal shares and measures is common in collaborative settings. Students might also appreciate challenges introduced through competitive games or through stories.
Session 1
In this session students begin to develop benchmarks for zero, half and one.
1/20, 6/10, 10/8, 11/12, 1/10, 3/8, 2/5, 9/10.
As the difficulty of this task depends on the fractions, begin with fractions that are clearly close to zero, half or one.
Why do you think 6/10 is close to half? How much more than a half is it? Why do you think 1/20 is close to zero? How much more than zero? Why is 11/12 close to one? Is it more or less than one? How much less?
3/10, 5/6, 5/9, 4/9, 18/20, 13/20, 2/8, 9/12, 1/5
Once more encourage the students to explain their decision for each fraction.
Session 2
In this session the students continue to develop their sense of the size of fractions in relation to the benchmarks of zero, half and one by coming up with fractions rather than sorting them.
5/6
How do you know that fraction is close to one?
5/6 7/8
How do you know that 7/8 is closer to one than 5/6?
Students might comment on how much needs to be added to each fraction to make it equal to one. "7/8 is closer to one because eighths are smaller parts than sixths and 7/8 is 1/8 smaller than 1 and 5/6 is 1/6 smaller than one. As 1/8 is smaller than 1/6, 7/8 is closer to one."
Do you notice a pattern as the fraction gets closer to one? (5/6, 7/8, 9/10, 11/12)
Which fraction is closer to one 99/100 or 999/1000? Why?
Which of these fractions is closest to zero, 1/3, 1/4, 1/5, or 1/6? Explain why?
Can a fraction have 5 as a numerator but still be close to zero? Give an example.
Session 3
In this session students estimate the size of fractions.
Ask for volunteers to record their estimate on the board. As they record estimates, ask each student to share their reasoning. Listen, without judgement to the estimate and then discuss why any given estimate might be a good one. Encourage students to share their justifications and ask questions of each other. There is no single correct answer but the estimates need to be reasonable. If the students are having difficulty, encourage them to reflect on the benchmarks developed in the previous sessions. Look for creative methods like estimating that the white area combines to 1/4 so the shaded area must be 3/4
Session 4
In this activity students identify which fraction of a pair is larger. The comparisons rely on an understanding of the top and bottom number (numerator and denominator) in fractions and on the relative sizes of the fractional parts. Equivalent fractions are not directly introduced in this unit but if they are mentioned by students they should be discussed.
2/5 or 2/8
Encourage explanations that show that the students understand that the fractions have the same number of parts but that the parts are different sizes. In this example 2/8 is smaller than 2/5 because eighths are smaller than fifths. Some students may know that 2/8 = 1/4. Use Fraction Strips to check students predictions.
4/5 or 4/6.
Encourage explanations that show that the students understand that both these fractions have the same count (numerator) but fifths are larger than sixths. Therefore 4/5 is greater than 4/6. Write 4/5 > 4/6. Use Fraction Strips to check students' predictions.
In this example the students can draw on their understanding of the benchmark of 1 and notice that 10/9 is larger than 1 and 9/10 is smaller than 1.
7/10 or 6/10
6/8 or 6/12
3/8 or 4/10 (more difficult)
9/8 or 4/3 etc
One fraction is greater than the other.
The two fractions have different numerators and different denominators.
What might the fractions be?
Session 5
In this session the students draw on their conceptual understanding of fraction benchmarks (0, 1/2, 1) and their understanding of the relative sizes of fractional parts to line fractions up on a number line.
Dear family and whānau,
This week we are learning about where fractions live on number lines. We have been especially interested in fractions that are close to zero, close to one half and close to one. Ask your child to put the following fractions on a number line and explain to you how they made up their mind about where to place them.
1/30 2/99 44/85 99/100 25/60 3/25 77/80 26/50