Write twenty-three thousand and forty. Explain the job of each zero.Write twenty-three thousand and forty. What do the zeros do?
Entry, 4 minutes. Mini-whiteboards; do not reveal until all have chosen and explained. Establish our goal: represent, compare and justify numbers, parts and rules.
Collect an individual response before revealing. Ask for the rule or diagram that supports it. Use the response to choose the next model or practice item.
Teacher answer: 23 040. The zeros hold the hundreds and ones places. 20 304 says twenty thousand three hundred and four.
Read the columns, then build a number with two ten-thousands, three thousands and four tens.Put 2 ten-thousands, 3 thousands and 4 tens in the table.
| Ten-thousands | Thousands | Hundreds | Tens | Ones |
|---|---|---|---|---|
| 2 | 3 | 0 | 4 | 0 |
20 000 + 3 000 + 40
Explicit model, 5 minutes. Place counters in columns. Empty places still need zero in the numeral. Guided: write twelve thousand and seventy; answer 12 070.
Collect an individual response before revealing. Ask for the rule or diagram that supports it. Use the response to choose the next model or practice item.
Teacher answer: 23 040. The digit 3 has value 3 000. Each place to the left is ten times the value of the place beside it.
Arrange the five numbers from smallest to largest. Which place settles each comparison?Put the numbers in order. Start at the left.
Ten-thousands → thousands → hundreds → tens → ones
5 minutes. Students arrange number cards. Then insert 9 050 between 9 009 and 9 090. Transfer to worksheet page 1 ordering task.
Collect an individual response before revealing. Ask for the rule or diagram that supports it. Use the response to choose the next model or practice item.
Teacher answer: 8 990 < 9 009 < 9 090 < 9 900 < 10 000. Compare equal place columns; the first different digit decides.
Someone says 14 507 is smaller than 14 480 because 0 is smaller than 8. Is the claim correct?Is 14 507 smaller than 14 480? Explain.
Compare the first place that differs, starting at the left.
5 minutes. Require the deciding place before selecting. First incorrect choice prompts retry; the second shows a place-value hint.
Collect an individual response before revealing. Ask for the rule or diagram that supports it. Use the response to choose the next model or practice item.
Teacher answer: The claim is incorrect. Ten-thousands and thousands match. Hundreds differ: 5 hundreds is greater than 4 hundreds, so 14 507 is greater.
Count the equal parts in each whole. How much is shaded?Each bar is one whole. Name the shaded part.
6 minutes. Fold equal-length paper strips or use printed strips. Unequal pieces do not represent these unit fractions. Guided: draw one fifth, then one tenth. Also name a half, a third and a quarter.
Collect an individual response before revealing. Ask for the rule or diagram that supports it. Use the response to choose the next model or practice item.
Teacher answer: One half, one third, one quarter, one fifth and one tenth. The denominator counts all equal parts; the numerator counts the shaded parts.
Two fifths are shaded. How much more must be shaded to complete the whole?Two parts are shaded. How many fifths are left?
5 minutes. Model counting the unshaded equal parts. Guided: seven tenths shaded; three tenths needed. Check students distinguish already shaded from still needed.
Collect an individual response before revealing. Ask for the rule or diagram that supports it. Use the response to choose the next model or practice item.
Teacher answer: Three fifths are needed. Two fifths plus three fifths makes five fifths: one whole. Keep the denominator 5 because the equal parts have not changed.
Compare one third and one fifth of equal wholes. Which part is larger? Explain with the bars.Same-size wholes: which is larger, one third or one fifth?
5 minutes. Invite an explanation before naming the rule. Ask whether the same conclusion is guaranteed for different-sized wholes; no. Then worksheet page 2.
Collect an individual response before revealing. Ask for the rule or diagram that supports it. Use the response to choose the next model or practice item.
Teacher answer: One third is larger than one fifth. Splitting the same whole into fewer equal pieces makes each piece larger. This comparison depends on equal-sized wholes.
Write sixteen thousand and twenty. Explain the value of 6. Draw three fifths and the part needed to make one whole.Write sixteen thousand and twenty. What is the 6 worth? Draw three fifths. Finish the whole.
Session 1 exit, 3 minutes, after 22 minutes of selected worksheet practice and feedback. Collect fresh evidence; reteach empty places or equal parts before session 2. This exit may be oral with a diagram.
Collect an individual response before revealing. Ask for the rule or diagram that supports it. Use the response to choose the next model or practice item.
Teacher answer: 16 020; the 6 is worth 6 000. Three of five equal parts shaded; two fifths are needed to complete the whole.
All racers complete the same course. Order their times from fastest to slowest, then rename 70 seconds.Same course: put the shortest time first. Rename 70 seconds.
Session 2 entry/model, 5 minutes. Group sixty counters then count the remainder; never treat 70 seconds as 0.70 minutes. Worksheet page 3 time task.
Collect an individual response before revealing. Ask for the rule or diagram that supports it. Use the response to choose the next model or practice item.
Teacher answer: 57 s, 59 s, 63 s, 70 s. 70 seconds = 1 minute 10 seconds; 63 seconds = 1 minute 3 seconds. Less time means faster for this same course.
Each player receives two cards. Predict the totals and explain the change when one more player joins.Two cards each. Find the totals. What does one more player add?
| Players | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Total cards | 2 | 4 | 6 | ? | ? |
5 minutes. Build groups of two counters for each player. Distinguish the input-output rule from adding 2 between successive totals. Worksheet page 3.
Collect an individual response before revealing. Ask for the rule or diagram that supports it. Use the response to choose the next model or practice item.
Teacher answer: 8 and 10 cards. Start with the number of players, multiply by 2, and record the total. One more player adds 2 cards; the total is not the number of players plus 2.
Choose five or ten cards per player. Write steps for any number of players, then test two and six players.Choose 5 or 10 cards each. Write a rule. Test 2 and 6 players.
4 minutes. Partners test each other’s instructions with a fresh input. A list of totals alone is not a general procedure. Worksheet page 3 rule task.
Collect an individual response before revealing. Ask for the rule or diagram that supports it. Use the response to choose the next model or practice item.
Teacher answer: Example: five cards each. Read the number of players; multiply it by 5; record the total cards. Two players need 10 and six need 30. One more player adds 5. For ten cards each: 20, 60 and an increase of 10.
Pack twenty-four counters into bags of three. How many bags? Connect the picture to two number sentences.24 counters. 3 in each bag. How many bags?
6 minutes. Treat each row as one bag; circle rows with the drawing tool. Then rotate the array to discuss different groups with the same total. Worksheet page 4.
Collect an individual response before revealing. Ask for the rule or diagram that supports it. Use the response to choose the next model or practice item.
Teacher answer: 8 bags. Eight groups of three is 24: 8 × 3 = 24. Twenty-four divided into groups of three gives 8: 24 ÷ 3 = 8.
Use four groups of seven to explain 28 divided by 4. Then describe the whole fact family.4 × 7 = 28. What is 28 ÷ 4? Why?
4 minutes. Contrast “how many groups?” and “how many in each group?” Draw both interpretations. Rehearse ×3, ×5 and ×10 related division facts on worksheet page 4.
Collect an individual response before revealing. Ask for the rule or diagram that supports it. Use the response to choose the next model or practice item.
Teacher answer: 28 ÷ 4 = 7. Four equal groups share the total 28, so each group has 7. Related facts: 7 × 4 = 28 and 28 ÷ 7 = 4.
Start at four. Record the number, add four, then stop if the new number is greater than 28. Otherwise record again.Start at 4. Record. Add 4. Is it more than 28? Stop if yes. Repeat if no.
6 minutes. Act out with current-number and recorded-list spaces on the board. Trace each decision, including the final unrecorded value. Worksheet page 5.
Collect an individual response before revealing. Ask for the rule or diagram that supports it. Use the response to choose the next model or practice item.
Teacher answer: Record 4, 8, 12, 16, 20, 24, 28. After recording 28, add 4 to get 32. Since 32 is greater than 28, stop without recording 32.
A program stops when the new number is greater than or equal to 28. Will it record 28? Explain what must change.Stop at “28 or more”. Does 28 get recorded?
4 minutes. Students trace rather than guess. Distinguish greater than from greater than or equal to. No need to teach a programming language.
Collect an individual response before revealing. Ask for the rule or diagram that supports it. Use the response to choose the next model or practice item.
Teacher answer: No. It stops after adding from 24 to 28, before recording 28. To include 28 with this order of steps, stop only when the new number is greater than 28.
Create steps that record the five-times-table numbers from five to twenty-five. Include a stop decision and test every step.Make a rule: 5, 10, ... to 25. Say when to stop.
5 minutes. Accept other valid orderings only after tracing them; e.g. test the current number before recording. Worksheet page 5 creates fresh steps independently.
Collect an individual response before revealing. Ask for the rule or diagram that supports it. Use the response to choose the next model or practice item.
Teacher answer: Start at 5; record the current number; add 5; if the new number is greater than 25, stop; otherwise return to record. Output: 5, 10, 15, 20, 25. The final tested value 30 is not recorded.
Compare the two recorded lists. Which numbers occur in both? Prove that your list is complete within these bounds.Find numbers in both lists. How do you know?
3 minutes. Students mark shared items, then explain the search. This does not require a formal lowest-common-multiple algorithm.
Collect an individual response before revealing. Ask for the rule or diagram that supports it. Use the response to choose the next model or practice item.
Teacher answer: 20 is in both lists. Check every item in the shorter list against the other list. We are comparing these finite lists; there are further shared multiples beyond them.
Rename 74 seconds. Use six players with five cards each. Write steps that record 3, 6, 9, 12 and stop.74 seconds = ? Six players, 5 cards each = ? Make steps for 3, 6, 9, 12.
3-minute exit after 10-minute selected independent practice/feedback. Check the endpoint and a fresh input. Use errors to plan another short lesson rather than assigning all unfinished items as a speed task.
Collect an individual response before revealing. Ask for the rule or diagram that supports it. Use the response to choose the next model or practice item.
Teacher answer: 1 minute 14 seconds; 30 cards. Start at 3; record; add 3; stop if the new number is greater than 12; otherwise return to record. Final tested value 15 is not recorded.
Select a slide to see accompanying pedagogical notes and teaching tips.