Collecting like terms, and what "simplest form" is for

What to assume they know

• Why every rule of arithmetic carries over to algebra — swapping, grouping, brackets and the distributive property, and the licence to use them on letters
• The dropped multiplication sign, and what it silently means — the dropped multiplication sign, so that 2l reads as 2 × l
• Perimeter of a rectangle, and area of a rectangle as length × breadth
• Adding and subtracting negative numbers

What they should be able to do

• Simplify an expression by collecting terms that carry the same letter-number
• State what makes two expressions equal, and check a claimed equality by substituting values
• Identify like terms and unlike terms in a given expression
• Remove a bracket that has a minus sign in front of it, inside an algebraic expression
• Expand a bracket that has a number in front of it, and collect the like terms that result
• Add and subtract two algebraic expressions
• Explain why an expression such as 18c + 11d cannot be simplified further
• Given a simplification, decide whether it is correct and, if not, repair it

Where it usually goes wrong

• "3a + 2b = 5, because 3 and 2 make 5." Item 1 of the p.94 block. The numbers are not free to be added: they are attached to different letters, and the sum would then have to be 5 of something. Ask which, and the error answers itself.
• "Simplest means shortest." 3(j + 2k + 3h + 4) is shorter than the expression it came from, and it is still not the simplest form the block asks for, because a bracket survives. The definition, not the length, decides.
• "Two expressions that look different are different." The chapter's whole method depends on the opposite. Example 11 reaches one expression by three routes on purpose, and the rectangle perimeter does it twice.
• "Checking at one value proves two expressions equal." It does not. The reason the perimeter forms are equal is that one was obtained from the other by rules that hold for all numbers — the substitution is a confirmation, not the argument. This distinction is exactly what §4.5 will turn into a proof about calendars.
• "A minus in front of a bracket flips only the first term." Item 5 of the p.94 block is this: 5 – (2 – 6z) claimed as 3 – 6z, where the correct form is 3 + 6z. Example 7 on p.91 is the repair: the minus reaches every term inside.
• "Subtracting means taking the smaller number from the larger." Item 4 of the same p.94 block is this, and it is a separate error worth its own name. In (4x + 3y) – (3x + 4y) each pair of front numbers has been differenced the comfortable way round — 4 – 3 for the x terms, and 4 – 3 again for the y terms — which is how the claimed x + y arises. The correct form is x – y. Flipping only the first term inside the bracket would have given x + 7y instead, so item 4 is not the misconception above wearing a different hat.
• "7p – p is 7p, because there is no number in front of the second one." Item 8. An unwritten 1 is still a 1, which is the mirror image of the unwritten multiplication sign in §4.3.
• "18c + 11d is unfinished work." It is finished. There is no rearrangement that turns two independent letters into one, and saying so out loud is more useful than leaving the student hunting.

Questions to check understanding

• Simplify a given expression — the p.103 list is the standard form of this question and runs to six items
• Add two given expressions; subtract one from another (pp.103, six items each)
• Say whether a claimed simplification is correct, explain the error and repair it
• Decide whether a given expression is already in simplest form, and justify
• Write an expression for a described situation and then simplify it
• Show that two given expressions are equal by simplifying both
• Describe a situation that a given expression could come from — the chapter asks this on p.85 and again on p.103

Examples worth working on the board

• Perimeter of a rectangle (Part I, §4.4, pp.87–88). Length l, breadth b, perimeter p. The long form is l + b + l + b; the section reorders it, pairs each letter with itself, and reaches 2l + 2b. The check the page runs is at l = 3 and b = 4, where both forms give 14. A blank rectangle is printed on p.87 immediately under the §4.4 heading, unlabelled; checked against p.87.
• Pencils and erasers (Example 5, Part I, §4.4, pp.88–89). A table with three day-columns. Pencils, at price c each: 5, 3, 10. Erasers, at price d each: 4, 6, 1. The page works the pencil row to 18c, states the combined expression 18c + 11d, and says it cannot go further because the two letters differ. Checked from p.88, where the row labels wrap inside their cells. It also asks the student to check that 5c + 3c + 10c and 18c agree at several values of c, and to take c = ₹50 in one case.
• Two rectangles from one (Example 6, Part I, §4.4, p.89). A rectangle of height v split by a vertical line into widths 4 and 3. Areas 4v and 3v; whole area either v × (4 + 3) or 4v + 3v. A side panel on the same page labels the figure as a second way to look at the distributive property. Checked from p.89 — the split rectangle and its dimension arrows are drawn art.
• Rectangle AEFD (Part I, §4.4, pp.89–90). ABCD has AB = 12 and height n; F and E cut off a strip of width 4 at the right, so AE = 12 – 4. The area of AEFD is 8n one way and 12n – 4n the other. The six vertex letters do come out of the text layer, but as two bare runs with the side lengths mixed in and nothing to say which corner is which, so the layout was read off the printed page of p.90.
• The sets that get named (Part I, §4.4, p.90). (5c, c, 10c) and (12n, –4n) are the page's examples of like terms; {18c, 11d} is its example of unlike terms.
• Chairs and tables (Example 7, Part I, §4.4, pp.90–91). Two small tables on p.90: rental per piece, chair ₹40 and table ₹75; amount returned, chair ₹6 and table ₹10. Renting x chairs together with y tables costs (40x + 75y) – (6x + 10y), and the page simplifies it to 34x + 65y, showing the grouping step (40 – 6)x + (75 – 10)y on the way. A Math Talk prompt then asks whether the same thing could have been written with the second bracket added rather than subtracted.
• Quiz scores (Example 8, Part I, §4.4, pp.91–92). Charu scores 7p – 3q in the first round, 8p – 4q in the second and 6p – 2q in the third, where p stands for the mark given when an answer is right and q for the penalty when it is wrong. At p = 4 and q = 1 the page works the first round to 25. The three rounds add to 21p – 9q. Krishita's total is given as 23p – 7q, and the page asks for the difference and for possible round-by-round scores that reach it, and asks what q would be if there were no penalty. It does not answer those.
• A number multiplying a bracket (Example 9, Part I, §4.4, p.92). The expression is 4(x + y) – y. The page names the distributive property, carries the 4 to each letter inside the bracket, rewrites the trailing subtraction as the addition of a negative term, and then folds the two y terms into one by writing their front numbers as 4 – 1, landing on 4x + 3y. It gives the explanation a worked model for carrying a number into a bracket that holds letter-numbers — the move items 3, 9 and 11 of the audit block ask for — and the unwritten 1 in front of the lone y is the same unwritten 1 that item 8 turns on.
• One picture, three counts (Example 11, Part I, §4.4, p.93). A block of four rows: four 3s, then r and s, then r and s again, then four 3s. Checked against p.93 — the 3s sit in green squares and the letters in coloured circles, none of which extracts in position. The page counts it three ways — row by row, like terms together, and upper half doubled — and all three simplify to 2r + 2s + 24.
• The eleven simplifications to audit (Part I, §4.4, pp.94–95). Each row gives an expression and beside it the form somebody has claimed is simplest, in this printed order:

1. 3a + 2b, claimed as 5

2. 3b – 2b – b, claimed as 0

3. 6(p + 2), claimed as 6p + 8

4. (4x + 3y) – (3x + 4y), claimed as x + y

5. 5 – (2 – 6z), claimed as 3 – 6z

6. 2 + (x + 3), claimed as 2x – 6

7. 2y + (3y – 6), claimed as –y + 6

8. 7p – p + 5q – 2q, claimed as 7p + 3q

9. 5(2w + 3x + 4w), claimed as 10w + 15x + 20w

10. 3j + 6k + 9h + 12, claimed as 3(j + 2k + 3h + 4)

11. 4(2r + 3s + 5), claimed as –20 – 8r – 12s

The block states what counts as simplest — brackets removed, like terms added, number-only terms added — and closes by asking whether the number of terms is related to the number of letter-numbers. The book prints no answers. Worked for this brief, only row 2 is already right; rows 9 and 10 are not wrong arithmetic but stop short of the stated finish.

Figures to have open

• The labelled rectangle for the perimeter argument, and the same rectangle again with 2l and 2b marked. Standard schematic.
• The split rectangle of Example 6, with widths 4 and 3 and height v. Standard schematic; the book's version carries a side panel naming it as the distributive property, which is worth keeping.
• Rectangle ABCD with E and F marked, AB = 12, the right strip 4 wide, height n. The vertex letters must be legible — they are the only way to say which rectangle is meant.
• The Example 11 block: four rows, 3s at top and bottom, r and s in the two middle rows. Redraw; the three counting routes should be possible to show over the same picture.
• The two small rental tables from p.90. Redraw; four cells each.
• No photograph is needed for this topic.

Where this sits in the book

• NCERT Ganita Prakash, Class 7, Part I, printed Chapter 4 "Expressions using Letter-Numbers", §4.4 Simplification of Algebraic Expressions, pp.87–95. Examples 5–9 and 11 sit on pp.88–93; the Figure it Out block is on pp.93–94; the unnumbered Mind the Mistake, Mend the Mistake block runs from p.94 to p.95.
• Example 10 and the Math Talk pair on pp.92–93 sit inside §4.4 but are notation arguments and are carried by The dropped multiplication sign, and what it silently means.
• Backward pointer: Part I, §4.2, p.85, for the distributive property, swapping and grouping — §4.4 uses all three and re-proves none of them.
• Forward pointer: Part I, §4.5, pp.98–99, where simplification stops being a tidying step and becomes the proof of a pattern.