Architecture of Grace
Mathematics · The Interior · Grades 6–8

The Interior Mathematics

Activity 14 · Last On, First Off. Solve it here, or print it and use a pencil — both count.

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Mathematics · Activity 14

Last On, First Off

Now x is wrapped twice — multiplied, then added to. Unwrap it like shoes and socks: whatever went on last comes off first, and whatever you do to one side, you do to the other.

On this computer only

No score is kept. No points, no streak, no ranking — the only number anyone writes down is the one your teacher writes down, standing next to you.

Show me one first

This one is nobody's problem — it is only here to show the moves before you touch yours. Watching it costs nothing and is written down nowhere.

2x + 5 = 9
Two wraps: x was multiplied by 2, then 5 was added.
Last on, first off: − 5 from both sides → 2x = 4.
÷ 2 on both sides → x = 2.

Now problem 1 is yours. The hint button is there the whole time, and using it is part of the lesson.

  1. 12x + 3 = 11
  2. 23x + 5 = 20
  3. 34x − 1 = 11
  4. 42x − 7 = 3
  5. 55x + 2 = 17
  6. 63x − 4 = −13
  7. 7x/2 + 3 = 8
  8. 8x/3 − 2 = 2
  9. 9−2x + 5 = 1
  10. 102x + 9 = 5
How confident are you with these?

There is no right answer here. It only tells your teacher where to start next time.

For the teacher — this device only

One probe row, built from what happened on this screen. Nothing is sent anywhere — copy it into the Goal Builder yourself, or print this page. Independent means solved with no hints; supported means solved after a hint. Both are real evidence and neither is a grade.

Date
—
Skill
Solve two-step equations
Measure
Trials (x of y)
Independent
0 of 10
Supported
0 of 10
Hints used
0
Fraction detours
0
Confidence
—

Items 1–6 are ax + b = c, 7–8 wear the division coat, 9 has a negative coefficient, 10 lands on a negative x. A fraction detour means the student divided before subtracting and met the fractions — not an error, but worth knowing: it is the reason the subtract-first strategy exists. Set 2 and Set 3 are the same ten skills with different numbers.

Saved on this computer