The Lumenary · Grades 3–5

One thing after another

Name: ______________________

Date: ______________________

You will learn to

Carry out chains of actions such as "add 5, then double" and predict their results
Find examples where swapping the order of two actions changes the result, and examples where it does not
Undo a chain of actions by undoing its steps in reverse order, as with socks and shoes, and explain why the original order fails
Draw arrow pictures showing how objects are transformed, and follow paths of arrows
Explain why four quarter turns in a row have the same effect as doing nothing

1.
Start with 4. Add 5, then double the result. What do you get?
2.
Start with 4 again. This time double first, then add 5. What do you get?
3.
For which pairs of actions does the order not matter? Choose all of them.
- A.Add 2 and add 7
- B.Double and triple
- C.Add 1 and double
- D.Put on socks and put on shoes
- E.Put on a hat and put on gloves
4.
Kiri puts on her socks, then her shoes, then ties her laces. How does she undo the whole chain?
- A.Take off socks, take off shoes, untie laces
- B.Untie laces, take off shoes, take off socks
- C.Take off shoes, untie laces, take off socks
5.
Wanjiru thought of a number. She multiplied it by 3 and then added 3, and got 27. What was her number?
6.
A spinner arrow can only turn clockwise, a quarter turn at a time. It has just been turned one quarter turn. How many more quarter turns bring it back to where it started?
7.
Musa turns a card 7 quarter turns clockwise, one after another. What is the smallest number of clockwise quarter turns that would leave the card in the same position?
8.
In the lesson's arrow picture, straight arrows mean add 1 and curved arrows mean double. What is the smallest number of arrows in a path from 1 to 7?
9.
Is there any starting number for which "add 5, then double" gives the same answer as "double, then add 5"?
- A.Yes: start at 0.
- B.No: "add 5, then double" always gives exactly 5 more.
- C.Yes, for a very big starting number.
10.
In Lean, andThen f g means "do f, then do g", and Undoes u f means "u undoes f". Suppose uf undoes f and ug undoes g. Which statement is the socks-and-shoes rule?
- A.Undoes (andThen ug uf) (andThen f g)
- B.Undoes (andThen uf ug) (andThen f g)
- C.andThen f g = andThen g f

Answer key — One thing after another

1.
$4 \xrightarrow{\text{add } 5} 9 \xrightarrow{\text{double}} 18$ . The answer is 18.
2.
$4 \xrightarrow{\text{double}} 8 \xrightarrow{\text{add } 5} 13$ . The answer is 13 — not 18. Swapping the order changed the result, by exactly 5, as the lesson proves.
3.
The order does not matter for add 2 and add 7 (both add 9), double and triple (both multiply by 6) and hat and gloves. It matters for add 1 and double (from 3: 8 versus 7) and for socks and shoes.
4.
Undo the steps in reverse order: the last step (tying laces) is undone first. So: untie laces, take off shoes, take off socks.
5.
Undo in reverse order. Undo "add 3": $27 - 3 = 24$ . Undo "multiply by 3": $24 \div 3 =$ 8. Check: $8 \to 24 \to 27$ ✓. (Undoing in the original order gives $27 \div 3 - 3 = 6$ , which does not work.)
6.
Four quarter turns make a full turn, which has the same effect as doing nothing. One has been done, so 3 more bring the arrow back.
7.
Four quarter turns do nothing, so 7 quarter turns have the same effect as $7 - 4 =$ 3 quarter turns. (That is a three-quarter turn — the same as one quarter turn anticlockwise.)
8.
One shortest path is $1 \xrightarrow{+1} 2 \xrightarrow{+1} 3 \xrightarrow{\times 2} 6 \xrightarrow{+1} 7$ : 4 arrows. Three arrows are not enough: after one arrow you can only be at 2; after two, at 3 or 4; after three, at 4, 5, 6 or 8 — never 7.
9.
No. "Add 5, then double" gives double the number plus 10, because the 5 is doubled too. "Double, then add 5" gives double the number plus 5. The first is always exactly 5 more, so they never agree. (Checking a few numbers suggests this; the doubling argument proves it for every number.)
10.
The rule is Undoes (andThen ug uf) (andThen f g): the chain "f, then g" is undone by "ug, then uf" — the undos in reverse order. In the lesson's Lean file this is socks_and_shoes, and wrong_order_fails shows that the original order does not work for "add 5, then double".