At a glance. One lesson of about 60 minutes. Students sort terms into like terms, combine them with their signs, and use substitution to check that the short form is equal to the original.
Calculator: no calculator. Nothing here needs one. The coefficients are small, and the focus is on structure, not arithmetic.
What students will learn
Know
- Like terms have the same letter part.
- A term keeps the sign in front of it.
- A letter on its own has 1 in front of it.
Understand
- Only like terms combine, because they count the same thing.
- Simplified means equal for every value of the letters, not just one.
- Terms with different letters stay separate.
Do
- Simplify expressions with one or two letters and plain numbers.
- Check a simplified form by substituting values.
- Spot and correct the mistake of adding unlike terms.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: collecting like terms accurately, including signs and lone letters (questions 2 to 8).
- Criterion B. Investigating patterns: the second figure. Ask students to find values of x and y where the wrong form happens to match, and to explain why matching at one pair is not enough.
- Criterion C. Communicating: explaining why 3x + 5y cannot be shortened and showing the substitution check (question 9).
- Criterion D. Applying mathematics in real-life contexts: a seller's stock in baskets and loose items, where only terms with the same letter combine, because x and y can change separately (question 5 and the hook).
The one thing to do with the figure
Ask students to sort before pressing Next step. Cover the figure and read out 3x + 5y + 2x − y. “Which terms belong together, and what does each pair make?”
Then use the second figure for one idea: the wrong form matches the original for some pairs of values, by chance, so one successful test never proves a simplification. Ask for a pair where it fails.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the first figure, with predictions first. |
| 10 min | Key idea; the second figure on checking by substitution. |
| 10 min | Worked example; students finish the faded one in pairs. |
| 5 min | Question 1, the check. If many choose A or C, return to the sorted row in the first figure. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 32; 22; 8. |
|---|---|
| Worked example | 7x + 4y + 4, which is 22 when x = 2 and y = 1. |
| Faded example | 4m + 7, which is 19 when m = 3. |
| 1. Collecting two letters | B, 6x + 4y. A joins all the terms; C drops the 1 in y; D mixes the coefficients of x and y. |
| 2. Like terms in one letter | 7. 5a + 2a = 7a. |
| 3. Collecting and substituting | 37. 3x + 7 with x = 10. |
| 4. Collecting two letters | 22. 4p + 5q with p = 3, q = 2. |
| 5. Collecting in a context | 54. 8b + 6 with b = 6. |
| 6. What cannot combine | C. A gives 3a, B gives 2y, D gives 2m + 2. Only 5a + 2b has no like terms. |
| 7. Terms that cancel | 30. 5x with x = 6. |
| 8. Working backwards | 5. 4 + 5 = 9. |
| 9. Unlike terms | B. A joins unlike terms; C turns y terms into x terms; D multiplies where the question adds. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Joined unlike terms, dropped the lone y, or mixed the coefficients. | Colour the x terms and y terms differently and collect each colour. |
| 10 or 3 in question 2 | Multiplied or subtracted the numbers in front. | 5 bags of a counters and 2 more bags of a counters is 7 bags of a: 7a. |
| 100 or 31 in question 3 | Treated plain numbers as x terms, or subtracted the numbers. | Sort into letter terms and plain numbers on two lines first. |
| 20 or 34 in question 4 | Lost the lone q, or added where the term was subtracted. | Write 1q beside the lone q, and keep each sign with the term after it. |
| 48 or 84 in question 5 | Left out the loose oranges, or multiplied them by the bag size. | Draw 8 bags and 6 loose ones, then count. |
| A, B or D in question 6 | Did not spot a pair of like terms. | List the letter part of every term first. |
| 54 or 42 in question 7 | Added where the term was subtracted, or ignored a term. | Underline each x term including its sign. |
| 9 or 13 in question 8 | Copied or added the plain numbers instead of finding the missing one. | Write the plain numbers on each side as an equation: 4 + k = 9. |
Questions worth asking
- Why can't 3x and 5y be added into one term? x and y are numbers that can change separately, so 3x + 5y has no single value of the form 8 of something. Listen for that, not ‘x is tomatoes’.
- Is 3x + 2x the same as 5x for every x? Yes. Test x = 1, 10 and 100: 5, 50, 500. That is what makes the short form valid.
- Does 5x − 5x give zero? Yes, 0x is nothing. The term disappears: 7p + 2q − 3p − 2q = 4p.
- Can unlike terms ever be equal? For particular values, yes, such as x = y = 0 in the second figure. Not for every value.
Differentiation
A way in for anyone stuck
- Use cubes or counters in two colours for x and y. Collect the colours in separate piles.
- Underline each kind of term with a different line (straight, wavy, boxed) before combining.
- Write every term with its sign and number, including 1x and 1y, before collecting.
Stretch
- Find a different expression that also simplifies to 5x + 4y and has exactly six terms. (Answers vary; each x pair must add to 5 and each y pair to 4.)
- Show that 7x − 3x + 2y is always an even number when x and y are whole numbers. (4x + 2y = 2(2x + y).)
Students learning in English as an additional language
The word term has an everyday meaning (a school term, a contract term). Show it on a written expression with each term circled. Like and unlike are also needed.
| Word | Plain meaning | Example |
|---|---|---|
| term | one part of an expression, with its sign | 3x and −y are terms of 3x + 5y + 2x − y |
| like terms | terms with the same letter part | 3x and 2x |
| unlike terms | terms with different letter parts | 5x and 4y |
| coefficient | the number in front of the letter | the 3 in 3x |
| simplify | write an expression in its shortest equal form | 3x + 2x simplifies to 5x |
| substitute | put a number in the place of a letter | put x = 2 into 5x |
Sentence starters for explaining aloud or in writing:
- The like terms are … and …, so they make …
- … and … are unlike terms because …
- I checked with x = …: the original is … and the short form is …
Two things not to say
“Add the numbers and join the letters.” That is exactly the 8xy error. Only like terms combine.
“Ignore the signs and sort them out later.” The sign belongs to the term after it; move the term and the sign moves too.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- Like termsSimplify 7k + 2k.
9k. - With a numberSimplify 10a − 4a + 3.
6a + 3.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextOne seller in Accra has 4 baskets of tomatoes and 5 loose ones. Another has 3 baskets and 8 loose. Each basket holds t tomatoes. Write the total and simplify it. How many is it when t = 12?
4t + 5 + 3t + 8 = 7t + 13, which is 97 when t = 12. - Earlier work, order of operationsWork out 5 + 4 × 3 − 8 ÷ 2.
13. 4 × 3 = 12 and 8 ÷ 2 = 4, then 5 + 12 − 4.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningSimplify 3x + 2y − x + 4y. Test your answer with two pairs of values. Why does the simplifying itself show they agree for every pair?
2x + 6y. For x = 2, y = 5 both are 34; for x = 1, y = 1 both are 8. Each step only regroups terms, so no value can change. - OpenWrite two terms that add to give 9x, and two terms that cannot be combined.
Answers vary, such as 4x + 5x = 9x, and 4x + 5y.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider or button, then the arrow keys or Enter) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Cut the terms out of paper and sort them on the desk.