Math Modelling: Planning a Maple Syrup Project

Remember from Grade 7

In Grade 7 you used the modelling process to choose a plan for a winter ski club. You made assumptions, wrote equations and inequalities, and revised your model when facts changed.

In Grade 8 you'll do it again with a bigger toolkit: linear patterns with rates and initial values, equations and inequalities with integers and decimals, and code that analyses data. The new challenge: real data will show that one of the assumptions was off, and you'll have to revise.

The situation

A Grade 8 class has permission to tap some sugar maple trees near the school, with an adult maple producer who will help. They want to make enough syrup for a pancake breakfast for 150 people in the spring.

Maple sap runs in late winter and early spring, when nights are below freezing and days are above freezing. For many First Nations in what is now Ontario, including Anishinaabe and Haudenosaunee communities, making maple sugar and syrup has been part of life for many generations, and many communities continue to gather at the sugar bush today. If a local community or Indigenous education centre shares teachings about the sugar bush, that is a great way to learn more.

How many taps does the class need, and how long will it take to collect enough sap?

1. Understand the problem

We know: 150 people are coming. Sap is boiled down to make syrup.

We need to find: how much syrup, how much sap, how many taps, and how many days.

We don't know: how much sap each tap will give. That depends on the trees and the weather.

2. Make assumptions

• Each person gets 30 mL of syrup (about 2 tablespoons).
• It takes about 40 L of sap to make 1 L of syrup. (This is a common rule of thumb. The real amount depends on how sweet the sap is.)
• Each tap gives an average of 2 L of sap per day over the season. (Some days give more, and cold days give none.)
• The class can put in 5 taps. (Tapping guides set a minimum trunk size for a tap, so not every tree can be tapped.)

3. Model and solve

Syrup needed: 150 × 30 mL = 4 500 mL = 4.5 L.

Sap needed: 4.5 × 40 = 180 L.

Sap collected: with 5 taps at 2 L each per day, the rate is 10 L per day and the initial value is 0. S = 10d.

Days needed: solve the inequality 10d ≥ 180, so d ≥ 18. The plan needs about 18 days of sap collecting.

Taps for a shorter season: if the season had only 12 good days, the class would need 2 × 12 × t ≥ 180, so 24t ≥ 180 and t ≥ 7.5. Taps come in whole numbers, so 8 taps.

4. Check and revise

After 10 days, the class has real data. They write code to analyse it:

Outputs: total = 85 L, mean = 8.5 L per day, "Behind plan: revise".

The assumption of 10 L per day was too high. The real rate is 8.5 L per day, which is 8.5 ÷ 5 = 1.7 L per tap per day.

Revised model. The class still needs 180 − 85 = 95 L.

• Option A: keep 5 taps. 8.5d ≥ 95, so d ≥ 11.2. They need 12 more days (about day 22 in total).
• Option B: add taps to finish by day 18 (8 more days). At 1.7 L per tap per day: 1.7 × 8 × t ≥ 95, so 13.6t ≥ 95 and t ≥ 6.99. They need 7 taps, which means adding 2 more.

But the sap season may end soon if the weather warms up. That's a risk to share.

5. Share

The class reports to the principal:

"We need about 180 L of sap to make 4.5 L of syrup for 150 people. After 10 days we have 85 L, which is behind our plan, because our taps are giving about 1.7 L a day each, not 2 L. We recommend adding 2 taps if the maple producer says the trees are big enough, so we can finish in about 8 more days. If not, we need about 12 more days of good sap weather. If the season ends early, we could serve smaller portions: 85 L of sap makes about 2.1 L of syrup, which is about 14 mL each for 150 people."