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Equivalent expressions
I worked these out from the practice questions. Each code says where to start reading, where to stop, what you can skip, and why it is safe.
1. Look at the options before you work
- Start:
- Look at the four options first.
- Stop:
- Stop once you see if they are products (factor) or sums (expand).
- Skip:
- Skip any work that does not lead toward that form.
- Why:
- Questions 3, 5 and 6 have products in the options, so factoring is the tool. Questions 1, 2 and 4 have sums, so expanding or combining is the tool.
Used in questions 1, 2, 3, 4, 5, 6
2. Test x = 2 (or another easy number)
- Start:
- Put an easy number into the original expression and get one number.
- Stop:
- Stop when only one option gives that same number.
- Skip:
- Skip all the algebra if one test already removes three options.
- Why:
- In questions 6 and 7 a single test number gives 5/6 and 5/3. For fractions, avoid numbers that make the bottom 0. Test two numbers if two options match.
Used in questions 1, 6, 7
3. Check the middle term
- Start:
- Find the options with the right first and last terms.
- Stop:
- Stop at the middle term sign and size.
- Skip:
- Skip the first and last terms, which the wrong options usually get right.
- Why:
- In question 5 the wrong options keep the right numbers but change the middle sign. In question 3 the (2x - 3)² trap adds a middle term.
Used in questions 3, 5
4. Brackets with a minus sign flip all signs
- Start:
- Find the subtraction in front of a bracket.
- Stop:
- Stop after flipping every term inside.
- Skip:
- Skip combining until the signs are fixed.
- Why:
- Question 4's wrong options each fail on one sign. Write the flipped line first.
Used in questions 2, 4
5. Fractions: factor then cancel, or common bottom
- Start:
- Look for the denominators.
- Stop:
- Stop when the top and bottom are factored, or the bottom is common.
- Skip:
- Skip cancelling anything that is added.
- Why:
- Questions 6 and 7 are the two fraction jobs. Only cancel factors, and never add tops and bottoms separately.
Used in questions 6, 7
6. Completing the square: half, square, subtract
- Start:
- Take half of the middle number.
- Stop:
- Stop when you have a and b.
- Skip:
- Skip the quadratic formula.
- Why:
- Question 8: half of 8 is 4, square it to 16, and subtract it from the constant. Vertex form (x + 4)² - 19 matches Desmos's vertex (-4, -19), a quick check.
Used in question 8
7. Table of values: test two numbers
- Start:
- Take the first two or three rows of the table and test each option.
- Stop:
- Stop when only one option matches every row you tested.
- Skip:
- Skip x = 0 if all options give the same value there.
- Why:
- In question 10 every option equals 5 at x = 0, and two options still match at x = 1. Only the second test number, x = 2, finishes the job.
Used in question 10
8. Exponent rules: multiply adds, divide subtracts, power multiplies
- Start:
- Square or cube everything inside the bracket first (the number too).
- Stop:
- Stop when each letter has one exponent.
- Skip:
- Skip expanding anything that is a single term.
- Why:
- Questions 11 and 13 are won by the same rules. In question 11 the trap is not squaring the 3. In question 13 a power 3/4 means root first, then power.
Used in questions 11, 13
9. Match the form the question asks for
- Start:
- Read the words after 'shows' or 'displays': zeros, maximum, or y-intercept.
- Stop:
- Stop when you know the form: factored (zeros), vertex form (maximum or minimum), standard form (y-intercept).
- Skip:
- Skip forms that are equivalent but do not show what is asked.
- Why:
- In question 14 two options are equivalent but only one shows the maximum. In question 16 the form a(x - h)² + k is vertex form, so complete the square.
Used in questions 14, 16, 8