Array for quadratics

It is expected that students will understand why the area of a rectangle is found by multiplying side lengths and that the product counts the number of units of area that fill the space. At level 5, students are also expected to have well-developed calculation strategies with whole numbers. Understanding and applying the distributive property of multiplication is particularly important.

1.

Begin by presenting some representations of area (perhaps of buildings or areas that are relevant to your students and your local context) and asking students to calculate the area. (Array for quadratics video 1) gives some examples for discussion. Encourage the students to use algebraic notation, e.g., 16 w rather than 16 x w.

Key points are:

• The factors multiplied are side lengths, but the product counts units of area (squares).
• Unknowns, like m and n, can be used to represent the area, even when the values are not known.
• Slides 7 and 8 provide a problem for investigation.
• How might the problem be written using algebra? (lw = 210)
• What values might l and w take up?
• Can either number be zero? Why not?

2.

Let the students investigate to find as many solutions as possible. Look for them to be systematic in organising the data. Suggest the use of a table if the data is not organised. You might model this on the board to get students started.

You may choose to plot the points using a graphing tool to see what patterns emerge.

Be open to students creating non-integral values in the ordered pairs, e.g., (1.5, 140), (28, 7.5). It is interesting to zoom in on the point defined by the ordered pair to check that it lies on the graph of lw = 210. Note that in this situation, both "l" and "w" are variables. This is worth pointing out to students.

3.

Provide students with (Array for quadratics activity 1). This asks them to find the value of the specific unknown using the given side length/s and the given area. Students may use mostly trial-and-error and improvement strategies. In doing so, they are likely to use estimation to determine plausible values and attend to the structure of each problem. Look for the following:

• Do students divide the area by the known side length to determine the side length that contains the missing value?
• In cases where the calculated side length is unknown, plus or minus a constant (e.g., j – 4 = 27), do students make the adjustment? Do they connect the plus or minus to the area diagram (e.g., j is four more than 27)?
• In cases where the unknown is on both sides, do they apply their knowledge of squares to estimate plausible values? (e.g., z (z – 1) = 182, so plausible values might be 13, 14, or 15).

4.

Gather together and review the strategies used, solutions, any misconceptions, and key learnings from the session.

In this session, students connect expressions with area representations. The expressions involve the distributive property of multiplication at different levels of complexity and the use of specific unknowns.

1.

Present the Array for quadratics video 2a and work through the problems given. 7 x 18 is used as a numeric example since students are most likely to apply the distributive property in the form of 7 x 10 + 7 x 8 or 7 x 20 - 7 x 2. The animation encourages students to write different expressions for the same way of seeing 7 x 18. The final slide suggests that arrays can be drawn as boxes, rather than needing to show every unit of area. Discuss the answers to the problems posed and address any misconceptions that arise.

2.

Ask the students to draw an array to represent 9 x 5 + 9 x 3.

• What other expressions could you write for the array? 

Look for 9 x (5 + 3) and 9 x 8.

3.

If students need more practice drawing arrays, provide other examples like (6 + 6) x 4 and 15 x (10 + 3). Show the Array for quadratics video 2b, which deals with numeric examples in which both factors are partitioned additively and progresses to the array representing (k + 7) (k + 9).

4.

Expect the students to recognise that the area can be represented with different equivalent expressions, such as:

24 x 16                  (20 + 4)(10 + 6)                  (20 x 10) + (20 x 6) + (4 x 10) + (4 x 6)

35 x 23                  (30 + 5)(20 + 3)                  (30 x 20) + (30 x 3) + (5 x 20) + (5 x 3)

(k + 7) (k + 9)      (k x k) + (k x 9) + (7 x k) + (7 x 9)                  k² + (9 + 7)k + 63

5.

Provide the students with the Array for quadratics activity 2 between two. This task requires students to match each array with two expressions and glue them in the same place in a workbook. The expressions are in factor and expanded form. Look to see if students make structural links such as:

• Look for the side lengths in the factor form.
• Attend to the partial products in the array pictures to find the expanded form.
• Recognise the conventions of recording multiplication, e.g., 7 x h as 7h, 3 x (6 + 4) as 3(6 + 4), and m x n as mn or nm.

In this session specific unknowns are introduced using array diagrams. Students are tasked with finding the value of those unknowns. 

1.

Display (Array for quadratics video 3) and wok through the problems as a class. This animation introduces simple problems in which the area of a square is known but the side length is not. Students may be aware that the square root function finds the side length of a square of given area, e.g. √49 = 7 so a square with sides of seven has area 49 square units. The example on Slide 5 represents the equation 11p = 264. Allow students to use estimation strategies if they wish but highlight the efficiency of dividing the area by 11 to find the value of p. 

2.

Use (Array for quadratics activity 3) (which contains a sequence of problems of increasing difficulty) to encourage students to use structur. Changes to the area make guess and check methods cumbersome. However, changes to the complexity of the side lengths increases the number of solution steps.

The problems are as follows. Students are expected to use their own strategies so the algebraic solutions are for your information.

8r = 840 → r = 105

15(m+4) = 300 → m + 4 = 20 → m = 16

w(w+4) = 437 → w² + 4w- 437 = 0 → (w + 23)(w - 19) = 0 → w = 19

h(h – 6) = 832 → h² – 6h – 832 = 0 → (h + 26)(h – 32) = 0 → h = 32

(k + 6)(k + 5) = 240 → k² + 11k -210 = 0 → (k + 21)(k – 10) = 0 → k = 10

(v + 7) (v – 3) = 459 → k² + 4k – 480 = 0 → (k + 24)(k – 20) = 0 → k = 20

In this session, the students use an array model to expand quadratic equations that are in factor form.

1. Display the Array for quadratics video 4a, which shows four examples of the form (x + a)(x + b). Work through the problems as a class, and be aware that students may need more practice to become fluent at expanding quadratics in factor form. (Array for quadratics video 4b) shows examples of the form (x + a)(x - b) or (x - a)(x + b).

Important points to address:

• Visualise the area that is represented by the factors.
• Consider how the terms in the factors will affect the area; e.g., negatives operating on positives create negative spaces.
• Tidy up the partial products by combining the x terms.

2.

Use the Array for quadratics activity 4 to provide students with more practice at expanding quadratic expressions in factor form.

This session is devoted to factorising quadratic expressions. 

1.

Work through the Array for quadratics video 5a together. This deals with the simplest form (x + a)(x + b) of factorising quadratic expressions. Ask students to complete the first four examples of (Array for quadratics activity 5) for practice. Expect students to connect their diagram with the algebraic factorisation, identify the important areas, and indicate whether the areas are positive or negative.

2.

Use the Array for quadratics video 5b to introduce the form (x + a)(x - b) and (x - a)(x + b). Work through the animation as a class. Students should practice examples 5-8 from (Array for quadratics activity 5) either cooperatively or independently. (Array for quadratics video 5c) deals with the form (x - a)(x - b). Work through the animation as a class. Examples 9–12 of the Array for quadratics activity 5 give practice factorising this form.

Solutions are as follows: