Rich tasks
- These are problem solving open-ended questions that require students to investigate fraction and decimal addition and subtraction. They can be given at the start of a topic and done as the students gain enough knowledge to complete them, inserted into a topic at an appropriate place, or given as homework, or extension.
- Banking on the interest
- A greedy pig
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1. Banking on the interest
- You could provide students with the latest bank interest rates and ask them in which bank they would put their money if they had $5000 to invest.
- Remind students that when interest is compounded six monthly, they work out the amount of interest the money would get for a year and then half it. This amount must then be added to the original amount invested. Then at the end of the year the interest is calculated on the new amount and halved and the interest added to the investment.
2. A greedy pig
- This will most likely be done using trial and error.
- Here is a more difficult version of the same problem.
- You could give students a clue and tell them that the denominator of the fraction for that day must be a factor of the number
of acorns the pig has at the beginning of that day. Otherwise he would get parts of acorns left and he always has whole numbers
of acorns.
This is another, more difficult, version of the problem.
A greedy pig was given 125 acorns.
Each day for six days he stashed a fraction of his acorns away and shared the rest with the other pigs. Then he ate one acorn from
his stash.
At the end of six days he had one acorn left.
The fractions he kept each day were 2/5, 17/25, 1/3, 5/6, 2/3 and 3/7.
Which fraction did he keep on each of days one to six?
Answer
Each day the pig kept a fraction of his acorns. The denominator of the fraction must be a factor of the number of acorns at the
beginning of the day and must not be the number 1. At the beginning of the first day, he had 125 acorns. The factors of 125 are
1, 5, 25, 125. Therefore on the first day, the only fraction(s) available are 2/5 and 17/25. Fractions for the next five days can be
eliminated in the same way.
The order of the fractions which the pig used are 17/25, 5/6, 2/3, 1/3, 3/7 and 2/5.
- Make up one of these problems yourself, starting with fewer than 100 acorns and choosing fractions so that there is one acorn left at the end of a week.
Fraction of
- Find a fraction of a whole number and multiply a whole number by a fraction and apply these skills to contextual problems.
- Find a fraction of a fraction and understand the algorithm for this.
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Teaching ideas
This link displays the getting started from page 107.
This reminds students about the meaning of fractions as parts of wholes.
Also included in this link is the Printable Master of pyramids for you to use.
This link displays 3/5 x 7 and the diagrams that go with it.
Students are asked to use diagrams to show how to calculate 2/3 x 8.
This link displays an example of six equivalent calculations.
Example 3/4 of 8 and 8 x 3/4 and (8 x 3)/4 and (3 x 8)/4 and 1/4 x 8 x 3.
Students are asked to write five equivalent calculations for four other fraction calculations.
This link displays diagrams for 2/3 x 3/4 and 2/5 of 3/8.
Asks students to shade them in.
You will need to display this on a write-on surface.
This link displays cards for you to print.
Give one set to each group.
Ask students to make them into sets of 4.
Example 2/3 x 5/8 and (2 x 5)/(3 x 8) and 10/24 and 5/12.
Teacher notes
- Emphasise that “of” can be replaced with x.
- Students sometimes multiply both the numerator and the denominator of the fraction by the whole number. Using materials or diagrams initially is helpful to student understanding.
- Diagrams and working with fraction pieces are good ways of helping with understanding fractions of fractions. Relating this to everyday life examples like Example 2 on page 7 is important as students can visualise the cake as well and understand how the diagram relates to it.
Decimal multiplication
- Solve problems that involve multiplying a decimal by a whole number and a decimal by a decimal, including decimals less than 1.
- Be able to estimate using decimals and know if an answer will be greater or less than the number being multiplied.
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Teaching ideas
- Revising multiplication and division strategies game
Choose a target number such as 256 or 288 or 600 and start at 2.
Divide the class into two groups. The groups take turns to multiply or divide the answer to the last calculation by a one-digit number. The winning group is the one that reaches the target number.
For example, 2 x 8 = 16 and 16 x 4 = 64 and 64 x 2 = 128 and 128 x 2 = 256.
This link displays Example 1 from page 115.
Students are then asked to fill in the gaps on some similar problems.
This will be best displayed on a write-on surface.
This link displays a problem that contains the multiplication 2•8 x 5.
Four options are shown for working out the answer to the problem with parts missing.
Ask students to choose one of the ways to complete.
Ask for volunteers to explain each strategy.
Ask students to discuss and then vote on which way is the most efficient.
Teacher notes
- A quick reminder of the main multiplication strategies of rounding and compensating, doubling and halving and using place value partitioning might be necessary. Discuss with students that some strategies work best with certain types of numbers.
- Students sometimes use the same strategy with every problem. Encourage them to choose strategies according to the numbers in the problem.
- Encourage students to estimate their answers first.
Dividing with decimals
- Solve problems that involve dividing a decimal by a whole number.
- Solve problems that involve dividing a decimal or whole number by a decimal amount.
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Teaching ideas
This link displays some decimal calculations done using either place value partitioning, rounding and compensating, or changing both numbers and asks students to fill in the gaps.
Discuss which is more efficient for number 2 and if there are any other strategies that could be used.
Ask if number 3 can be done in any other way, and if so which way is more efficient.
Discuss how students can decide which strategy to use.
This link displays a range of calculations with decimals and asks students to choose which is the correct equivalent calculation.
Teacher notes
- Students sometimes have trouble grasping the idea of doing an equivalent calculation when dividing by a decimal. Remind them that this is the same reasoning as is used for the strategy of changing both numbers. Example 64 ÷ 16 changed to 32 ÷ 8 is similar reasoning to changing 1.6 ÷ 0.4 to 16 ÷ 4. In the first example both numbers are halved and in the second both numbers are multiplied by 10. The key is that the same thing has been done to both numbers so that the answer doesn’t change.
- Encourage students to estimate their answer first.
Mixed multiplying and dividing decimals
- Solve a mixture of problems by multipying or dividing decimals.
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Teaching ideas
You could use any of the teaching ideas not already used from the two previous sections.
Teacher notes
- Students need to decide whether to multiply or divide when solving problems. Looking at the language and context helps.
Percentage of
- Find percentages of quantities using mental, written or calculator methods.
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Teaching ideas
This link displays a weta with 6 legs to fill in.
On Sunday the weta walks 50 m.
How far does he walk on the other days if he walks these percentages of 50 m?
50%, 40%, 25%, 45%, 72%, 125% of 50 m
You will need to display this on a write-on surface.
You could use this as a class discussion.
- Activity 10 question 1 on page 127
You could use this as a class activity.
You will need to display this on a write-on surface.
Teacher notes
- Students will need to be proficient at finding fractions of amounts and also know the fraction/decimal equivalents for percentages such as 10%, 50%, 25%, 33 1/3%, 12 1/2% etc.
- Encourage students to estimate their answers.
- Remind students to put units with their answers.
- GST and tax are a good example of the use of percentages. Remind students that GST is added to the cost of an item and tax is taken off someone’s income.
Fractions, decimals and percentages
- Use multiplicative strategies in a range of situations with fractions, decimals and percentages.
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Teaching ideas
You could use any of the teaching ideas in the previous sections that you haven’t already used.
Teacher notes
- It is important that students make the connections between fractions, decimals and percentages and realise that all three can be used to describe the same proportion.
- Once students understand what fractions, decimals and percentage are, they will find converting between them much easier. Learning algorithmic ways of converting between them is not helpful.
Find the missing number
- Find the number when a percentage or fraction of the number is given.
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Teaching ideas
Give students 3 counters each.
Tell them this story.
The counters you have represent three students.
These three students are the quarter of students in a gym class who could do a somersault.
How many students are in the gym class?
Ask students how they worked out the answer was 12.
Repeat with these.
Give 4 counters and tell students this is 1/3 of the group.
Give 8 counters and tell students this is 50% of the group.
Give 5 counters and tell students this is 20% of the group.
Give 4 counters and tell students this is 40% of the group.
Teacher notes
- Students will understand this concept better if it is demonstrated with materials first.
- Students often confuse this with fraction or percentage of.
- Encourage students to find what a unit fraction is. If they are told what 3/5 is they first must find what 1/5 is and then 5/5 (one whole).
If they are told what a percentage is that is not a factor of 100 they must first find a percentage that is a factor of 100. If they are told what 60% is then they must first find what 10% is and then 100% (one whole).
(Getting started 6)