Exploring Algebraic Identities

20 small steps. Take them one at a time.

1. 1Step 1.

   Predict first

   A square garden is 10 m on each side. It is made 3 m longer in both directions, so it becomes 13 m by 13 m. Predict: how many square metres are added?

   Show what happensWhat happens

   1. Strip along one side: 10 × 3 = 30
   2. Strip along the other side: 10 × 3 = 30
   3. Corner where they meet: 3 × 3 = 9
   4. Added: 30 + 30 + 9 = 69

   Answer 69 m²: two strips of 10 × 3 = 30 m² each, and a corner of 3 × 3 = 9 m².

   Common answers, and what they can mean

   • 9: The corner is new, but so are two long strips along the sides, each 10 m by 3 m. The next steps show all three.
   • 60: The two strips are right: 30 + 30. Where they meet there is also a small 3 by 3 corner.
   • 39: The garden grows in two directions, so there are two strips, and a corner where they meet.
   • 169: 169 m² is the whole new garden. The added part is 169 − 100.
2. 2Step 2.

   Picture a square grid with 5 small squares on each side. Cut each side into a part of 3 and a part of 2. The grid breaks into four pieces.

3. 3Step 3.
   Whole grid:5 × 5 = 25

   Four pieces:9 + 6 + 6 + 4 = 25

   Count the small squares in two ways. The four pieces fill the grid exactly.

   Nothing is lost and nothing is extra.

4. 4Step 4.

   Investigate: Square builder

   The sliders start at a = 10 and b = 3, like the garden. Here a is the long part and b is the short part. A square of side a has area a × a, written a². Look at the four pieces. Then move b and watch which pieces grow.

   CheckPartly done for you

   Which piece is the new corner of the garden?

   Show the answerAnswer

   Answer The small square, b².

   CheckPartly done for you

   How many a by b strips are there?

   Show the answerAnswer

   Answer Two.

   Common answers, and what they can mean

   • One: There is one strip along the top and one down the side: two strips, each a × b.

   Your turnPartly done for you

   Use a and b to write the area of the whole square as the sum of its four pieces. Here a and b are more than 0 (a > 0 and b > 0), because they are lengths.

   Hint 1Hint 1

   There is a big square, two strips and a small square.

   Hint 2Hint 2

   a² for the big square, ab for each strip, b² for the corner.

   Show the answerAnswer

   1. Big square: a²
   2. Two strips: ab + ab = 2ab
   3. Corner: b²
   4. Whole: a² + 2ab + b²

   Answer a² + 2ab + b²

   Common answers, and what they can mean

   • a² + b²: a² and b² are there, but the two ab strips are part of the square too.
   • a² + ab + b²: There are two strips, so ab appears twice: 2ab.
5. 5Step 5.

   Your turn

   Take a 6 by 6 grid. Cut each side into a part of 4 and a part of 2. How many small squares are in each piece?

   Show a hintHint

   Look for a 4 by 4 square, two 4 by 2 strips and a 2 by 2 square.

   Show the answerAnswer

   16, 8, 8 and 4.

   1. Together they make 36, which is 6 × 6.
6. 6Step 6.
   Big square:a × a = a²

   Two strips:a × b + a × b = 2ab

   Small square:b × b = b²

   Now use letters for any size. Call the long part a and the short part b, so each side is a + b. They are lengths, so both are more than 0 (a > 0 and b > 0). The whole square is (a + b)², which means (a + b) × (a + b).

   ab + ab is 2ab, just as 6 + 6 is 2 × 6.

7. 7Step 7.
   (a + b)² = a² + ab + ab + b²

   = a² + 2ab + b²

   Each part you add, like a², ab or b², is called a term — one part of a sum. Add the pieces to get the whole square. Writing (a + b)² as its pieces is called expanding — opening the brackets and writing out every part.

   expand: open the brackets and write out every part

   term: one part of a sum, like a², ab or b²

8. 8Step 8.

   Explain why

   Why is the middle term 2ab, and not ab?

   Show the answerAnswer

   Answer There are two strips, and each one is a × b.

   Common answers, and what they can mean

   • Because the power is 2: The ² means a square, not 'times 2'. The 2 in 2ab counts the strips: one along the top, one down the side.
   • Because a is added to b twice: Look at the picture: the 2 counts two strips, each a × b.
9. 9Step 9.

   Watch out

   Wrong(a + b)² = a² + b²

   Right(a + b)² = a² + 2ab + b²

   The wrong rule forgets the two strips. With 3 and 2 it gives 9 + 4, which is 13. But the grid has 25 small squares.

10. 10Step 10.

    a and b	(a + b)²	a² + 2ab + b²
    3 and 2	25	9 + 12 + 4 = 25
    −4 and 1	9	16 − 8 + 1 = 9
    1/2 and 1/2	1	1/4 + 1/2 + 1/4 = 1
    5 and −5	0	25 − 50 + 25 = 0

    A side cannot be −4 long, so the picture needs positive numbers. Does the rule work for negatives and fractions? The table tests some. Both sides match in every row.

11. 11Step 11.

    Check

    Suppose b = −3. What about the square picture and the rule (a + b)² = a² + 2ab + b²?

    Show the answerAnswer

    Answer The picture cannot show it (a side cannot be −3 long), but the rule is still true.

    Common answers, and what they can mean

    • The rule stops working: Test it: with a = 7 and b = −3, (7 − 3)² is 16, and 49 − 42 + 9 is 16. The next steps show why the rule works for every number, even where the picture cannot go.
    • The picture works, and so does the rule: The rule works, but the picture cannot: no side can be −3 long. The next steps show another way to prove the rule.
12. 12Step 12.
    (a + b)² = (a + b)(a + b)

    = a(a + b) + b(a + b)

    You cannot test every number, so multiply the brackets out. Think of the second bracket as one number, like a box. The distributive property says: multiply each part inside a bracket by what is outside.

    Like 3(x + 2) = 3x + 6: the 3 goes with each part.

    distributive property: multiply each part inside a bracket by what is outside

13. 13Step 13.
    a(a + b) + b(a + b)

    = a² + ab + ba + b²

    = a² + 2ab + b²

    a(a + b) means a × a plus a × b, so it is a² + ab. In the same way, b(a + b) is ba + b². Here a and b are variables — letters that stand for numbers that can change.

    ab and ba are the same, like 3 × 2 and 2 × 3.

    variable: a letter that stands for a number that can change

14. 14Step 14.
    x + 4 = 9

    x = 5: 5 + 4 = 9(true)

    x = 6: 6 + 4 = 10(not 9)

    Some equations are true for only some numbers. x + 4 = 9 is true only when x is 5. An identity is a special equation — a rule that works for every number you try. So the rule for (a + b)² is an identity.

    identity: a rule that works for every number you try

15. 15Step 15.
    (a + b)² − (a² + b²) = 2ab

    (a + b)² and a² + b² differ only by 2ab. If 2ab is positive, (a + b)² is bigger. If it is negative, (a + b)² is smaller. If a or b is 0, they are equal.

16. 16Step 16.

    Your turn

    Test the rule with a = 7 and b = −3. Do both sides match?

    Show a hintHint

    Square a + b first. For 2ab, plus × minus gives minus.

    Show the answerAnswer

    Yes, both give 16.

    1. Left side: 7 − 3 is 4, and 4² is 16.
    2. Right side: 7² is 49 and (−3)² is 9.
    3. 2 × 7 × (−3) is −42.
    4. So 49 − 42 + 9 is 16.
17. 17Step 17.
    13² = (10 + 3)²

    = 10² + 2 × 10 × 3 + 3²

    = 100 + 60 + 9

    = 169

    The identity helps you square numbers in your head. Split the number into a round number and a small one.

    So Meera's grid has 169 small squares.

18. 18Step 18.

    Your turn

    Find 31² in your head.

    Show a hintHint

    Write 31 as 30 + 1.

    Show the answerAnswer

    961.

    1. That is 900 + 60 + 1.
19. 19Step 19.
    (5m)² = 5m × 5m

    = 5 × 5 × m × m

    = 25m²

    The letters a and b are like empty slots. Any term, like 5m, can go in a slot. Square the whole term.

20. 20Step 20.

    Watch out

    Wrong(5m)² = 5m²

    Right(5m)² = 25m²

    Square the number as well as the letter. When m is 2, (5 × 2)² is 100. But 5 × 2² is only 20.