At a glance. One lesson of about 60 minutes. It assumes Lessons 1 and 2. Students find the first whole period at which a model passes a target by a table, read a crossing time from a GDC graph, decide between exact and whole answers, and compare an exponential with a line.
Calculator: GDC optional. A GDC graph and its intersection tool give the crossing time directly, which is the Diploma habit. The table by hand comes first so that students know what the intersection means; logarithms are not needed here.
What students will learn
Know
- The time is in the exponent, so trial and a table of values find it.
- The first whole period is the exact crossing rounded up (the crossing itself if it is a whole number).
- The GDC's intersection gives an exact decimal time.
Understand
- Rounding to the nearest is wrong for a first-time-reached question.
- A growth curve is below the target before the crossing and above after it.
- A rough rule gives an estimate of the crossing, not the whole period.
Do
- Make a table of a model for t = 0, 1, 2, ….
- Read the first whole period for a target.
- Use a graph to find a crossing and interpret it.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: a table of values, the first whole period, reading a crossing (questions 1 to 7).
- Criterion B. Investigating patterns: the second figure. Ask students to find how the doubling year changes with the rate and to test the rule of 70.
- Criterion C. Communicating: writing a table with a conclusion in words and a check by substitution.
- Criterion D. Applying mathematics in real-life contexts: users of an app, doubling time, savings plans, a falling value.
The one thing to do with the figure
Make students commit before they slide. In the first figure, ask “Which period will the curve first be above the target?” for a base of 1.5 and a target of 500 before moving the target.
In the second figure, ask what the doubling year would be for 10% a year and compare with the rule 70 divided by the rate. Ask why the rule gives 7 and the answer is 8.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: waiting for a thousand. Collect predictions of more or less than 10 weeks. |
| 10 min | First figure and key idea: table first, then the crossing. |
| 10 min | Second figure, the warning card and the worked examples; pairs finish the faded example. |
| 5 min | The check. If many choose A, ask which row of the table is the first at or above 100. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace, with the GDC for those who want to check. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 40; 1.15; 3.375. |
|---|---|
| Hook | 30 × 1.410 is about 868 and 30 × 1.411 is about 1215, so the app first passes 1000 users in week 11. |
| Worked example | 20 × 1.64 = 131.072, so t = 4; 900 × 0.65 = 69.98, so t = 5, with the crossing at 4.3. |
| Faded example | 91.125 at t = 4 and 136.6875 at t = 5; t = 5. |
| 1. First whole period from a table | B. A is the last t below the target; C gives the y-value at t = 5; D gives the target. |
| 2. First whole period, growth | 5. 405, then 1215. |
| 3. First whole period, decay | 7. 12.5, then 6.25. |
| 4. First whole year to double | 7. 1.9738 then 2.2107. |
| 5. Whole period from a GDC crossing | 7. The next whole number after 6.1. |
| 6. Time to fall below half | 5. 626.4 then 532.4. |
| 7. Exponential against a line | 6. 320 against 200. |
| 8. Interpreting a crossing for decay | C. A rounds to the nearest; B gives the crossing; D has the right number for a rising curve. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Gives the last t below the target, the y-value, or the target. | Ask: which column of the table is t and which is y? |
| 4, 1215 or 200 in question 2 | Stops one short, gives y, or gives the quotient. | Ask: is 405 more than 1000? Is 1215? |
| 6, 6.25 or 80 in question 3 | Stops one short, gives y, or gives 800 ÷ 10. | Write both rows: 12.5 and 6.25, and ask which is below 10. |
| 6, 8 or 2.21 in question 4 | Takes 1.97 as double, uses a simple-interest rule, or gives the value. | Ask: is 1.9738 more than 2? |
| 6, 6.1 or 5 in question 5 | Rounds to the nearest, gives the crossing, or gives the quotient. | Substitute t = 6: is 193.1 at least 200? |
| 4, 3 or 600 in question 6 | Takes 626.4 as 600, takes 15% of the start off each year, or gives the target. | Ask: is 626.4 below 600? |
| 5, 4 or 320 in question 7 | Stops one week short, compares with Q's start, or gives the value. | Make a two-column table of both plans. |
| A, B or D in question 8 | Rounds to the nearest, gives the crossing, or thinks the curve rises. | Ask: is the base above or below 1? Then substitute t = 3. |
Questions worth asking
- Why can you not just divide the target by the start? The time is in the exponent, so the quotient is the size the power must reach, not the time.
- What does 4.3 mean for the model y = 900 × 0.6t and the target 100? The value falls to 100 part of the way through the fifth period, so the first whole t below 100 is 5.
- When is rounding to the nearest wrong? When the question asks for the first period at or beyond the target.
- Does a bigger base always reach a target sooner? Yes, for a start below the target, because the values are larger at every period after 0.
Differentiation
A way in for anyone stuck
- Give a table with the t column filled and the y column to complete by multiplying.
- Use a checklist: write the table; mark the row below the target and the row above; read t.
- Let students use a repeated-multiplication shortcut on the calculator (the previous answer times the base).
Stretch
- A model y = 6 × 1.25t has to pass 100. Without a table, find two whole numbers between which t must lie, using powers of 1.25.
- Show that 70 divided by the rate in percent is a good estimate of the doubling time for 5% but a poor one for 50%.
Students learning in English as an additional language
“Target” is a number you want to reach. “Crossing” is where a curve meets a line. “First whole period” means the first whole number of steps after which the value is at or beyond the target. “Intersection” is the point where two graphs meet.
| Word | Plain meaning | Example |
|---|---|---|
| target | a value that you want a quantity to reach | a stock of 500 items |
| table of values | a list of t and y for t = 0, 1, 2, and so on | the rows of a model by hand |
| crossing | the point where the curve meets the target line | the time at which the curve reaches the line |
| first whole period | the smallest whole t at which the value reaches the target | the first week with at least the target |
| intersection | a point where two graphs meet | found with a GDC tool |
| doubling time | the time for a quantity to become twice as big | the years for savings to double |
Sentence starters for explaining aloud or in writing:
- At t = … the value is …, which is below / above the target.
- The first whole period is … because ….
- The GDC crossing is at …, so the first whole t is ….
Two things not to say
“Round the answer to the nearest whole number.” Look at the value on both sides of the crossing.
“Divide the target by the start to get the time.” That gives a size of the power, not the time.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- Tabley = 2 × 3t. Find the first whole t with y at least 50.
3. y = 54 at t = 3; 18 at t = 2. - Decayy = 160 × 0.5t. First whole t with y below 20.
4. 160 → 80 → 40 → 20 → 10.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA town's population is y = 8000 × 1.05t. In which whole year does it first exceed 10 000?
5. 1.054 = 1.2155, 1.055 = 1.2763. - Earlier work (Lesson 2)A quantity of 100 mg has a half-life of 3 hours. How much is left after 9 hours?
12.5 mg.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningThe GDC says a curve crosses 90 at t = 2.04. Is the first whole t 2 or 3? Say what you would check.
3, because at t = 2 the value is just below 90; check by substitution. - Working backwardsy = a × 2t first exceeds 100 at t = 5, but not at t = 4. Give a whole-number a that works.
a = 4 gives 64 at t = 4 and 128 at t = 5. Any a with 100 ÷ 32 below a and a ≤ 100 ÷ 16 works: a = 4 to 6.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
All sliders work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each figure, “Show me the steps” gives the same states in words.
No device? Give each pair a two-column table template and a calculator with a repeated-multiplication shortcut.