MAE201M/201/1/2016

Tutorial letter 201/1/2016

Pre-Algebra and Algebra Education in

Intermediate and Senior Mathematics

MAE201M

Semesters 1

Department of Mathematics Education

Feedback on Assignment 2

QUESTION 1 [25]

1.1 What is meant by algebraic thinking? (2)

Responses may vary

Algebraic thinking can be interpreted as an approach to quantitative situations

that emphasizes the general relational aspects with tools that are not necessarily

letter- symbolic, but which can ultimately be used as cognitive support for

introducing and for sustaining the more traditional discourse of school algebra.

(Kieran, 1996, p. 275)

1.2 Write a paragraph on the meaning and importance of algebraic thinking in (2)

intermediate and senior phase mathematics.

Responses may vary

To explore properties and relationship of numbers, as well as operations on

numbers

To discourage memorization of rules or properties; to allow students to analyse

many simple and specific cases that help them move beyond thinking about

multiple specific instances to thinking about underlying mathematical

generalization.

1.3 Nono takes a number and subtracts 9 from it. He takes the same number and (2)

subtracts 12 from it. He then takes these two new numbers and multiplies them. The

result he gets is 0. (3)

(x – 9)(x – 12) = 0 (2)

1.3.1 x2 21x + 108 = 0

1.3.2 either x – 9 = 0 or x – 12 = 0

x = 9 or x = 12

1.3.3 9 or 12

1.3.4 Responses may vary (3)

To demonstrate and meaningfully show the importance of simplifying and

recognizing equivalent expression algebraically. Provide meaning and to make

sense to solve for x in an algebraic equation.

1.4 You want to introduce the concept of a function to grade 8 learners.

You set the following problem to your learners:

Mpho is trying to make money to help pay for his studies at the technikon by selling

hot dogs from a hot-dog cart at a soccer stadium during major games. He pays the

cart owner R75 per night for the use of the cart. He sells hot dogs for R12,50 each. His

costs for the hot dogs, napkins, etc. are R5,60 per hot dog on average. Calculate the

number of hotdogs that Mpho has to sell to make a profit.

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1.4.1 Write down the four steps of Polya’s problem solving process. MAE201M/201

1. UNDERSTANDING THE PROBLEM (4)

2. DEVISING A PLAN

3. CARRYING OUT THE PLAN

4. LOOKING BACK

1.4.2 Responses may vary (7)

QUESTION 2 [32]

2.1 What is meant by "an algebraic expression"? (2)

Responses may vary

An expression consisting of arithmetic numbers, letters (used as symbols) and

operation signs

2.2 List five different ways to develop algebraic reasoning in your learners. (10)

a) Generalisation from arithmetic and from patterns in all of mathematics

b) Meaningful use of symbolism.

c) Study of structure in the number system.

d) Study of patterns and functions.

e) Process of mathematical modelling, which integrates the first four.

2.3 The number 100 can be written as the sum of consecutive whole numbers like this:

100 = 18 + 19 + 20 + 21 + 22

2.3.1 Is it possible to find a sequence of any number of consecutive whole numbers (3)

which add up to 1000?

198 + 199 + 200 + 201 + 202 = 1000

2.3.2 Can every number be written as a sum of consecutive whole numbers? (3)

No, any number that is a power of 2 can’t be written as a sum of

consecutive numbers.

2.3.3 Is there a general method of writing down such a sum? Explain –Yes (2)

2.4 (4)

2.4.1 Find four consecutive odd numbers whose sum is 120:

27 + 29 + 31 + 33 = 120

2.4.2 A pawpaw tree when planted was 90 cm tall. It then grew an equal number of

centimetres each year. At the end of the seventh year it was taller than at

the end of the sixth year. How tall was the tree at the end of the twelfth year?

T12 = ? [T7 T6 = ] (8)

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T1 = 90 cm; d = ;

Tn = a + (n 1)d

T12 = 90 + 11 = 91.22 cm

QUESTION 3 [25]

A ball is thrown into the air. Its height (in metres) at time t (in seconds) is given by the (6)

function h(t) =9t – t².

3.1 Complete the following table to find the height of the ball after 6 seconds.

Time (s) Height (m) h

0 0

1 8 20

2 14 18

3 18

4 20 14

5 20

6 18 8

0 2 4 6 8 10 t (4)

3.2 Plot the points on a system of axes and connect them with a smooth curve.

3.3 Use your graph to answer the following questions: (1)

(a) At what times is the golf ball on the ground? t = 0s & t = 9s (1)

(b) At what time is the ball at its highest point? t = 4,5 s (1)

(c) How high does the golf ball go? h = 20.25 m

3.4 Plan an activity on number patterns for the intermediate phase. (5)

The activity should address

Solve problems that involve grouping and sharing.

Investigate and extend numeric (to at least 5 000)and geometric patterns

looking for general rules or a relationship, including patterns:

descriptions of the same relationship or rule represented:

- Verbally;

- in flow diagrams;

- by number sentences.

Write down the instructions you will give the learners. It must be clearly stated (2)

what learners must do.

3.5 Design a rubric that you will use to assess the learners’ answers. (4)

A rubric should at least exhibit: (1)

Models of the performance or product.

Establishment of the criteria.

Number of levels in the rubric.

Descriptions of quality for each level of the criteria.

3.6 What resources/learning support materials will you use?

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Generic resources: MAE201M/201

Board

Learner book (DoE) [25]

Pictures

Writing book

QUESTION 4

4.1 Consider the adjacent diagram.

4.1.1 If four congruent square corners are cut off from the large square of cardboard (5)

to make a box, write down the area of the shaded figure in terms of x and y. (5)

(5)

Area of the shaded region = y2 - 4x2

= y(y - 2x) + 2x(y - 2x)

= (y - 2x)(y - 2x) + 4x(y - 2x)

4.1.2 Use this result to find this area when y = 11.8 cm and x = 1.4 cm

Area of the shaded region = (11.8)2 – 4(1.4)2

= 131.4 cm2

4.1.3 The sides are folded up to form a box. Draw the box and indicate all its

dimensions.

4.1.4 Write down an expression in terms of x and y for the volume of the box. (5)

(5)

Volume = (y - 2x)( y - 2x) x

4.1.5 Use the values of x and y from your answer to question 4.1.2 and determine the

volume of the box.

Volume = lbh = (11.8-2*1.4)(11.8-2*1.4)(1.4) = 113.4 cm3

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