Birthday cakes

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Achievement Objectives
NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
Specific Learning Outcomes

Find fractions of a set using multiplication and division.

Description of Mathematics

Number Framework Stage 6

Required Resource Materials
Counters (to represent candles)

Paper circles (to represent cakes)

Activity

Background Maths

Fractions involve a significant mental jump for the students because units of one,which are the basis of whole-number counting, need to be split up (partitioned) andrepackaged (re-unitised). It is crucial that the students have significant opportunitiesto split up ones through forming unit fractions with materials and are required torecombine several of these new units to form fractions like two-thirds and five quarters.

In this way, the students are required to co-ordinate the link between the numerator
and denominator in fraction symbols.
Early additive students are progressing towards multiplicative thinking. Fraction
contexts offer opportunities for these students to appreciate the links between addition
and multiplication. This can be achieved through requiring the students to anticipate
the results of equal sharing and involving fractions like thirds, fifths, and tenths,
where addition methods are less efficient.
It’s important that you require the students to construct the whole unit from given
parts. For example, if a student is given a Cuisenaire rod or pattern block and told that it’s one-quarter of a length or shape, they should be able to reconstruct the whole.
Similarly, given two red cubes and being told that the cubes are one-fifth of a stack,
students can make the whole stack.
Understanding the relationship between fractions, as numbers, and the number one is
critical for further learning. Ensure that the students know that a whole, as referred to
in fraction work, means the same as the number one.
 

Using Materials

Problem: “Four people are at Carla’s birthday, so they will get one quarter
(one-fourth) of the cake each. Carla has 16 candles to put on the cake so that each
person gets the same number of candles on their piece of cake. How many candles
will each person get on their piece of cake?”
Give the students 16 counters and a paper circle to fold into quarters.
Ask, “How many candles do you think each person will get?” “How do you know?”
Look for the students to use adding-related strategies, such as halving and halving
again, like “Eight and eight is 16, so one-half of the cake has eight candles. Four and
four is eight, so one-quarter has four candles.”
Confirm the answers by equally sharing the counters onto the paper circle. There is noneed to do this if the students are able to use partitioning strategies.
Pose similar problems like:
“Five people are at the party. There are 25 candles.”
“Three people are at the party. There are 21 candles.”
Record the answers using symbols, for example, 1/5of 25 is 5.
Vary the problems by using non-unit fractions, like, “At the party the cake is cut into
quarters (fourths). Twelve candles are put on the cake.
Greedy Greg eats three-quarters of the cake. How many candles does he get?” Recordthe results using symbols, that is, 3/4 of 12 is 9.
 

Using Imaging

 Problem: (Show the students one-fifth  of a paper circle with four counters on it.)

 
 cakes1.
 
 
“Here is a piece of Rongopai’s birthday cake. Each piece of cake has the same
number of candles. How old is Rongopai?”

Reconstructing: Listen to the students’ ideas about how many candles were on the whole cake. Fifths have been deliberately chosen because they are easily confused with quarters. Ask the students to justify their answers. 

“I think the cake was cut into five pieces.”

“What would each piece be called?” (one-fifth)
“If there are five pieces, then there are 4 + 4 + 4 + 4 + 4 candles on it.
That’s 20.”
“How did you work out that was 20?”
The whole cake can be constructed, if necessary, to confirm the answers.
Record the answer as: 1/5 of ?  is 4, so ? is 20. This means that Rongopai is 20 years
old.
 cakes2.
 
Pose similar problems and ask the students to work out the number of
candles on the whole cake.
1/3 of ? is 6
3/4 of ?  is 9
3/5 of  ? is 6
 

Using Number Properties

Ask similar word problems and record them using symbols.

For example:
“Two-thirds of the cake has eight candles on it. How many candles are on the whole
cake?” 2/3 of ? is 8, so ? is 12.
“Three-quarters of the cake has nine candles on it. How many candles are on the
whole cake?” 3/4 of  ? is 9, so ? is 12.
 

Independent Work

The students will benefit from playing the games Chocolate Chip Cheesecake (see Material Master 7-1, and Mystery Stars Material Master 7-8)

 

 
 
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Level Three