Lesson plan Grade 7 Math
Pinebrook's Two-Stage Water Rate Increase: A Grade 7 Percent Lesson
A 50-minute grade 7 lesson where students apply two successive percent increases to a water bill and compare the result to simply adding the two percentages.
Objectives
- Students will calculate the result of two successive percent increases applied to the same starting amount.
- Students will compare a compound percent increase to the simple sum of the two individual percentages and explain why the results differ.
Vocabulary
Key words: percent increase, compound increase, base amount
Teacher overview
Pinebrook Mutual Water Company (a fictional water utility used for this lesson) has approved a rate increase that takes effect in two stages: an 8 percent increase in the first year, followed by a 5 percent increase in the second year, calculated on the already-increased amount. This lesson uses one household's monthly bill to walk students through both stages and asks them to compare the compound result to what they would get by simply adding 8 percent and 5 percent together. A common misconception in this lesson is assuming that two successive percent increases of 8 percent and 5 percent combine to a flat 13 percent increase; because the second increase is calculated on a larger base amount, the true combined increase is slightly more than 13 percent, and the procedure is built to make that difference visible with real numbers.
Procedure
Hook (5 min): Ask students to predict, without calculating: if a $54 bill from Pinebrook Mutual Water Company (a fictional utility used for this lesson) goes up 8 percent one year and then 5 percent the next year, what is the combined percent increase over the two years? Collect a few guesses on the board, including any that say "13 percent," without correcting them yet.
Direct instruction (15 min): Model the two-stage calculation using the Whitfield family's Pinebrook water bill of $54.00 per month. Show Stage 1: $54.00 × 1.08 = $58.32. Show Stage 2, applied to the new amount: $58.32 × 1.05 = $61.24 (rounded to the nearest cent). Calculate the total dollar increase ($61.24 − $54.00 = $7.24) and the true combined percent increase ($7.24 ÷ $54.00 ≈ 13.4 percent), then return to the board guesses and discuss why 13.4 percent is larger than the naive 13 percent sum.
Guided practice (15 min): In pairs, students repeat both stages for a second Pinebrook household, the Bellamy family, whose current bill is $72.00 per month, using the same 8 percent and then 5 percent increases. Pairs check their Stage 1 and Stage 2 answers against each other before moving on.
Independent practice (10 min): Each student answers two questions on the handout:
- What is the Bellamy family's total dollar increase across both stages?
- What is the Bellamy family's true combined percent increase, compared again to the naive 13 percent sum?
Closure / exit ticket (5 min): Students answer one question on an index card: "A third Pinebrook household has a $40.00 bill. After both stages, is the new bill closer to $45.20 (the naive 13 percent estimate) or a slightly higher amount? Explain in one sentence why." Collect and scan for the correct reasoning (that the second increase compounds on the larger Stage 1 amount) before dismissal.
From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.
Answer Key
- Answer: $9.65.
Stage 1: $72.00 × 1.08 = $77.76. Stage 2: $77.76 × 1.05 = $81.65 (rounded to the nearest cent). Total increase: $81.65 − $72.00 = $9.65. - Answer: about 13.4 percent.
$9.65 ÷ $72.00 ≈ 0.134, or about 13.4 percent, the same combined rate as the Whitfield example, because compounding 8 percent and 5 percent always produces the same overall multiplier (1.08 × 1.05 = 1.134) no matter the starting bill. - Answer: $45.36.
This is the exit-ticket answer, for a third household's $40.00 bill. $40.00 × 1.08 × 1.05 = $45.36, which is higher than the naive combined estimate of $45.20 ($40.00 × 1.13) because the Stage 2 increase of 5 percent is applied to the already-larger $43.20 Stage 1 amount, not to the original $40.00.
Differentiation
Support (below grade level)
Provide a two-row calculation table with column headers "Starting amount," "× 1.08," and "× 1.05," so students only need to fill in the multiplication results rather than set up the structure themselves. Allow a calculator for every step.
Extension (above grade level)
Ask students to find the single multiplier that represents both stages combined (1.08 × 1.05 = 1.134) and use it to calculate a fourth household's bill in one step rather than two. Then ask them to find what single-stage percent increase would produce the exact same result as the two combined stages.
ELL supports
Pre-teach "base amount," "increase," and "combined" using a simple diagram showing a bar growing in two steps. Provide sentence starters for the exit ticket: "The new bill is ___ because the second increase is applied to ___."