Back to the on-screen lesson ·
Working out what has to happen first.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you solve problems that need two steps, deciding what has to be worked out before the question can be answered. That deciding is the whole lesson: the arithmetic is arithmetic you already have, and what is new is holding a plan in your head — do this first, then that — which is a genuinely different skill from calculating.
You can add when parts join and subtract when a part is removed or a gap is measured. A two-step problem uses those familiar actions together. The challenge is to find an amount that the question does not give directly. Read the whole problem before calculating. Tell someone what happens in your own words. You may move counters, draw boxes, or write a short note after each event. These records help you remember what each number means while you decide what to do next.
| Term | What it means |
|---|---|
| known | An amount or relationship the problem gives. |
| unknown | The amount you are trying to find. |
| intermediate result | An amount found along the way and used in the next step. |
| equation | A statement that two amounts have equal value. |
| change | In a money problem, the amount returned after paying more than the price. |
Suppose you have 28 beads, receive 15 more, and use 19 to make a bracelet. The question asks how many remain. You cannot remove the 19 from the original 28 and forget the gift. First find the amount after the gift: 28 + 15 = 43. Then remove the beads used: 43 - 19 = 24. The number 43 is an intermediate result. It answers a helpful smaller question, not the final question.
Label that result: 43 beads before making the bracelet. The label tells you why the next step starts there. A bare 43 written in a corner is easy to misuse. The two equations also tell the story in order. Keep the equals sign honest: do not write 28 + 15 = 43 - 19 = 24, because 43 is not equal to 24. Write separate equations, or write 28 + 15 - 19 = 24 after you understand the two actions.
Another way: table
| Point in the story | Number of beads |
|---|---|
| Before the gift | 28 |
| After the gift | 43 |
| After making the bracelet | 24 |
Each row names a different moment; it is not a list of three answers to the same question.
A story can give a starting amount, a change, and an ending amount, but the unknown may be any one of them. If 18 children are in a room, 7 arrive, and 25 are there afterward, addition finds the ending amount. If the story instead says some children were there, 7 arrived, and now 25 are there, the start is unknown. Subtracting 7 from 25 recovers the 18 who were already there. The word arrived appears in both stories, yet the useful calculation changes.
This is why a word such as more does not always tell you to add. If Kai has 16 shells, which is 5 more than Mira has, Mira has 11. You compare the larger amount with the extra part to find the smaller amount. Draw one bar for Kai and a shorter bar for Mira. The extra piece of Kai's bar is five. Removing that extra piece from sixteen leaves the amount matching Mira's bar.
For a two-step problem, ask what the final question names and which missing fact would help answer it. You might need a combined price before finding change, an earlier total before removing a part, or one person's amount before combining it with another person's amount. Write a box for each unknown. A plan should say what each box means. The plan is useful even before you know the numbers that belong in the boxes.
A classroom has 72 markers. One group takes 18 and another group takes 24. You can follow the events: 72 - 18 = 54, then 54 - 24 = 30. You can also combine the amounts removed first: 18 + 24 = 42, then 72 - 42 = 30. Both ways remove the same two separate groups of markers. The grouping is helpful because it makes the shared action clear: both groups take markers away.
Do not replace the second subtraction with addition just because you have reached a new sentence. The story, not the line break, tells you what happens. Likewise, do not subtract the first group twice. Point to each amount in your calculation and name the event it represents. A short checklist can help: first group accounted for, second group accounted for, starting collection used once.
The two methods have different intermediate results. Fifty-four is the amount left after the first group. Forty-two is the total taken by both groups. They do not need to be equal because they answer different smaller questions. The final remainder is the same. When comparing methods, compare their meanings rather than expecting every line to contain the same number. This is one way to explain why two different-looking solutions can both be correct.
In our pretend classroom store, an eraser costs 26 cents and a pencil costs 39 cents. A child pays with one dollar, which is 100 cents. First add the prices: 26 + 39 = 65 cents. Then find the change: 100 - 65 = 35 cents. The first calculation names money spent; the second names money returned. If you stop at 65, you have found the cost but have not answered the change question.
Keep all three amounts in cents while calculating. Subtracting 65 from the digit 1 would mix cents and dollars. The number 1 stands for one whole dollar here, not one cent. Converting the payment to 100 cents makes the units match. Write cents or the cent symbol beside the answer so that 35 cannot be mistaken for thirty-five dollars. The dollar sign belongs with an amount in dollars.
Change can also be found by counting up from the total cost to the payment. From 65 to 70 is 5 cents, from 70 to 100 is 30 cents, and 5 + 30 = 35. This checks the subtraction. The total cost and change together should equal the payment. Our prices are invented and have no extra charge. If a real purchase includes another charge, it must be included in the cost before computing change. Never assume the story gave a price that it did not state.
Consider this story: a club receives 14 cards, then gives away 9, and has 37 cards left. How many did it start with? The ending amount is given, so following the events forward from an unknown start is awkward. Undo the last event first. Before giving away nine, the club had 37 + 9 = 46 cards. Then undo the earlier arrival of fourteen: 46 - 14 = 32. The starting collection had 32 cards.
Working backward does not mean reversing every digit or subtracting everything. It means undoing each action in reverse order. To undo giving away, put that amount back. To undo receiving, remove the amount received. Each intermediate result needs a time label: forty-six before the gift away, thirty-two before the delivery. Without those labels it is easy to use the right operations in the wrong order.
Now test the result by telling the story forward. Start with thirty-two. Receive fourteen to reach forty-six. Give away nine to reach thirty-seven, exactly the stated ending amount. This forward check is particularly useful for missing-start problems. It tests whether the proposed start fits all events, not merely whether one subtraction was calculated correctly. If the last amount does not match, return to the first event that disagrees and repair that part of the plan.
A problem might say that a box is blue, has a label numbered 8, and holds 45 crayons. If it asks how many crayons remain after 12 are used and 7 more are used, the label number is not part of the count. Forty-five, twelve and seven describe crayon amounts. Eight identifies the box. Adding every number in the paragraph would answer no meaningful question. Read the units and roles, not just the digits.
After calculating, use a check that matches the story. For two removals, put the removed groups back with the remainder. For a delivery followed by a removal, add the removed amount back to the end and compare it with the starting amount plus the delivery. For a missing-start story, run the story forward. These checks use relationships between quantities; they are stronger than merely repeating the same keystrokes.
Estimate the answer's direction too. If a collection gains fifteen and then loses nineteen, it should end four below its start. In the bead story, twenty-four is four below twenty-eight. This is a quick comparison, not a substitute for understanding the two events. If you lose track, make a fresh table with one row per moment. Write the original facts first and leave the unknown rows empty until you work them out. A tidy record can reveal whether the arithmetic or the plan needs repair.
At a school math event, a pretend store in Iowa lists a folder for 31 cents and a paper bookmark for 18 cents. A visitor pays 75 cents in play money. First calculate the combined price: 31 + 18 = 49 cents. Then calculate the change: 75 - 49 = 26 cents. Check that 49 + 26 = 75. If a helper returns 44 cents after subtracting only the folder price, the bookmark has not been included. You can explain the mistake by pointing to the missing part of the cost, rather than just saying the answer is wrong. The situation uses invented prices and play money. Its mathematical purpose is to coordinate two costs and one payment using the same unit.
A teacher prepares 80 counters for games. One table receives 23 and another receives 29. How many remain for a third table? Following the deliveries gives 80 - 23 = 57 and 57 - 29 = 28. Combining the deliveries first gives 23 + 29 = 52 and 80 - 52 = 28. The intermediate numbers differ because they describe different quantities: fifty-seven after the first delivery, fifty-two delivered altogether. Both methods leave twenty-eight. Before putting materials away, the class checks 23 + 29 + 28 = 80. If the actual count differs, they recount the groups and compare their records. The calculation checks the plan; it does not prove that nobody dropped a counter while moving the boxes.
If a question asks for change, a correct total price is only part of the solution. If it asks for the start, the stated ending amount is not the answer. Circle the words that name what is unknown, then label your answer with those words. Another trap is using a keyword rule: more always means add, or left always means subtract. A missing-start problem can require addition to undo a loss. Look at how the quantities are related. Finally, separate equal statements from successive events. The equals sign means equal value; it does not mean and then. Keep each equation true even when the story has several changes.
Name the starting collection.
28 beads.
This is the amount before either event.
Find the amount after delivery.
28 + 15 = 43 beads.
The gift joins the collection.
Remove the beads used.
43 - 19 = 24 beads.
The bracelet takes beads out of the collection.
Answer the final question.
24 beads remain.
The question asks about the ending amount, not the delivery total.
Check the net change.
19 - 15 = 4; 28 - 4 = 24.
The loss exceeds the gain by four.
Choose one money unit.
1 dollar = 100 cents.
Payment and prices must use matching units.
Combine the two prices.
26 + 39 = 65 cents.
Both items must be paid for.
Find the unspent part.
100 - 65 = 35 cents.
Change is the payment minus the total cost.
Check by counting upward.
5 + 30 = 35 cents from 65 to 100.
The gap can be filled in convenient jumps.
Rebuild the payment.
65 + 35 = 100 cents.
Cost and change are the two parts of the payment.
State the requested amount.
The change is 35 cents.
Sixty-five cents names the cost instead.
Name the final collection.
37 cards after receiving 14 and giving away 9.
The starting amount is missing.
Undo the last event.
37 + 9 = 46 cards.
Return the cards given away.
Undo the earlier event.
46 - 14 = 32 cards.
Remove the cards received to recover the original collection.
Label the recovered amount.
The club started with 32 cards.
This answers the unknown-start question.
Replay the delivery.
32 + 14 = 46.
The proposed start must fit the first event.
Replay the giveaway.
46 - 9 = 37.
It must fit the second event too.
Compare with the stated finish.
37 = 37.
The complete story agrees with the proposed start.
Find the total removed.
17 + 26 = 43 tickets.
Both events reduce the same collection.
Find the remainder.
64 - 43 = 21 tickets.
Remove the combined group once.
Check the partition.
Label the final answer.
Nia had 34 stickers, then got 7 more, then gave 9 away. How many stickers are there now?
Answer:
Two prices are 25 cents and 14 cents. A payment is 100 cents. Find the change.
Find the total cost.
Add the two prices before comparing with the payment.
Both purchases use part of the money.
Calculate the amount returned.
The change is change cents.
Subtract the combined cost from the payment.
Rebuild the whole payment.
The two costs and the change must total one hundred cents.
The payment is partitioned into spent and returned money.
A shop had 85 apples. It sold 11 in the morning and 21 in the afternoon. How many apples are left?
Answer:
Omar had 45 stickers, then got 10 more, then gave 16 away. How many stickers are there now?
Answer:
A shop had 64 apples. It sold 27 in the morning and 22 in the afternoon. How many apples are left?
Answer:
Ines had 47 stickers, then got 17 more, then gave 12 away. How many stickers are there now?
Answer:
Match each event and plan to what it does to Kofi's number of cards.
| add 13 | take away 9 | finds the number of cards now | |
|---|---|---|---|
| buys 13 more cards | |||
| gives 9 cards to a friend | |||
| 17 + 13, then take away 9 |
At an invented classroom store, a pencil costs 27 cents and a folder costs 12 cents. A customer buys both and pays 78 cents. With no other charge, how many cents should be returned?
Answer:
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A number increases by 24, then decreases by 9, ending at 42. Complete the two missing stages. Work backward, then check forward.
| Before the decrease | Original number | |
|---|---|---|
| Missing stages |
You can solve a problem that takes two steps. Without looking: in a problem where you buy two things and work out the change, what has to happen first?
16. Your turn: start with 64 tickets, give away 17, then give away 26, step 3
21 + 17 + 26 = 64.
Remaining and removed tickets rebuild the start.
16. Your turn: start with 64 tickets, give away 17, then give away 26, step 4
21 tickets remain.
The intermediate total of forty-three is not the remainder.